Original Article Journal Section Sara Neves Silva, King’s College London,
London, UK<EMAIL_ADDRESS>
# Fully automated planning for anatomical fetal brain MRI on 0.55T

Sara Neves Silva  Centre for the Developing Brain, School of Biomedical
Engineering & Imaging Sciences, King’s College London, London, UK  Biomedical
Engineering Department, School of Biomedical Engineering & Imaging Sciences,
King’s College London, London, UK  Sarah McElroy  Centre for the Developing
Brain, School of Biomedical Engineering & Imaging Sciences, King’s College
London, London, UK  MR Research Collaborations, Siemens Healthcare Limited,
Camberley, United Kingdom  Jordina Aviles Verdera  Centre for the Developing
Brain, School of Biomedical Engineering & Imaging Sciences, King’s College
London, London, UK  Biomedical Engineering Department, School of Biomedical
Engineering & Imaging Sciences, King’s College London, London, UK  Kathleen
Colford  Centre for the Developing Brain, School of Biomedical Engineering &
Imaging Sciences, King’s College London, London, UK  Biomedical Engineering
Department, School of Biomedical Engineering & Imaging Sciences, King’s
College London, London, UK  Kamilah St Clair  Centre for the Developing Brain,
School of Biomedical Engineering & Imaging Sciences, King’s College London,
London, UK  Biomedical Engineering Department, School of Biomedical
Engineering & Imaging Sciences, King’s College London, London, UK  Raphael
Tomi-Tricot  Centre for the Developing Brain, School of Biomedical Engineering
& Imaging Sciences, King’s College London, London, UK  MR Research
Collaborations, Siemens Healthcare Limited, Camberley, United Kingdom  Alena
Uus  Centre for the Developing Brain, School of Biomedical Engineering &
Imaging Sciences, King’s College London, London, UK  Biomedical Engineering
Department, School of Biomedical Engineering & Imaging Sciences, King’s
College London, London, UK  Valéry Ozenne  CNRS, CRMSB, UMR 5536, IHU Liryc,
Université de Bordeaux, Bordeaux, France  Megan Hall  Biomedical Engineering
Department, School of Biomedical Engineering & Imaging Sciences, King’s
College London, London, UK  Department of Women & Children’s Health, King’s
College London, London, UK  Lisa Story  Biomedical Engineering Department,
School of Biomedical Engineering & Imaging Sciences, King’s College London,
London, UK  Department of Women & Children’s Health, King’s College London,
London, UK  Kuberan Pushparajah  Biomedical Engineering Department, School of
Biomedical Engineering & Imaging Sciences, King’s College London, London, UK
Mary A Rutherford  Centre for the Developing Brain, School of Biomedical
Engineering & Imaging Sciences, King’s College London, London, UK  Biomedical
Engineering Department, School of Biomedical Engineering & Imaging Sciences,
King’s College London, London, UK  Joseph V Hajnal  Centre for the Developing
Brain, School of Biomedical Engineering & Imaging Sciences, King’s College
London, London, UK  Biomedical Engineering Department, School of Biomedical
Engineering & Imaging Sciences, King’s College London, London, UK  Jana Hutter
Centre for the Developing Brain, School of Biomedical Engineering & Imaging
Sciences, King’s College London, London, UK  Biomedical Engineering
Department, School of Biomedical Engineering & Imaging Sciences, King’s
College London, London, UK  Smart Imaging Lab, Radiological Institute,
University Hospital Erlangen, Erlangen, Germany

###### Abstract

Purpose: Widening the availability of fetal MRI with fully automatic real-time
planning of radiological brain planes on 0.55T MRI.
Methods: Deep learning-based detection of key brain landmarks on a whole-
uterus EPI scan enables the subsequent fully automatic planning of the
radiological single-shot Turbo Spin Echo acquisitions. The landmark detection
pipeline was trained on over 120 datasets from varying field strength, echo
times and resolutions and quantitatively evaluated. The entire automatic
planning solution was tested prospectively in nine fetal subjects between 20
and 37 weeks. Comprehensive evaluation of all steps, the distance between
manual and automatic landmarks, the planning quality and the resulting image
quality was conducted. Results: Prospective automatic planning was performed
in real-time without latency in all subjects. The landmark detection accuracy
was 4.21$\pm$2.56 mm for the fetal eyes and 6.47$\pm$3.23 for the cerebellum,
planning quality was 2.44/3 (compared to 2.56/3 for manual planning) and
diagnostic image quality was 2.14 compared to 2.07 for manual planning.
Conclusions: Real-time automatic planning of all three key fetal brain planes
was successfully achieved and will pave the way towards simplifying the
acquisition of fetal MRI thereby widening the availability of this modality in
non-specialist centres.

###### keywords:

Fetal MRI, motion detection, motion correction, tracking, fetal brain
development, T2* relaxometry

## Introduction

MRI plays an increasing role in both clinical antenatal diagnosis and
research, complementing ultrasound (US) screening for a range of suspected
fetal pathologies. The most common clinical indication [1] for fetal MR
imaging are thereby suspected brain anomalies. MRI has demonstrated specific
additional utility when compared with US in evaluating the posterior fossa
[2], midline structures, the cortex in the progressively ossifying fetal
skull, evidence of haemorrhage [3], cysts, cleft palates [4] and head and neck
tumours [5], among others.

MRI offers higher spatial resolution, enhanced soft tissue contrast, and a
wide range of functional contrasts. However, it poses unique challenges, such
as safety considerations, involuntary fetal motion, artifacts arising from
air-tissue boundaries and variability in fetal position and maternal
surroundings. The most widely used acquisition technique is T2-weighted 2D
single-shot Turbo Spin Echo (ssTSE), providing excellent contrast and an in-
plane resolution of 1-1.5 mm, while effectively freezing motion for each
individual slice.

Radiological assessment requires high-quality images in a set of defined fetal
brain planes to perform a set of key measurements including the bi-parietal
diameter, trans-cerebellar diameter, and ventricle diameters among others, and
to visualize essential structures like the corpus callosum [6] and cerebellar
vermis to identify deviations from normal development. Furthermore,
gyrification and cortical development may be visualised in detail. Given the
size of the studied structures, the accuracy of these measurements is crucial
and relies on the availability of exact sagittal, coronal, and axial slices.
While recent advances in Slice-to-Volume-Reconstruction (SVR) [7, 8, 9, 10]
allow 3D reconstructed high-resolution 3D volumes and thus re-orientation to
true brain anatomy in cases where sub-optimal, oblique native planes are
acquired, this technique is currently only available in specialist centres and
often performed offline. Specialized radiographers trained to perform this
challenging planning for fetal MRI are essential. Typically, acquisition
planning is performed by optimizing the angles manually and iteratively on
stacks acquired in different orientations such as whole uterus sagittal and
coronal ssTSE sequences.

Another recent development is the rediscovery of low-field 0.55T MRI. It
addresses some of the aforementioned challenges of fetal MRI: the increased
field homogeneity reduces distortion artifacts, the larger bore size widens
access to pregnant women with larger body mass index and in later gestation,
the reduced heating allows more efficient acquisitions and the longer T2* is
beneficial for widely performed T2* relaxometry measurements. Recent studies
showed its benefits as a promising tool for fetal MRI [11, 12, 13]. The
reduced cost, footprint, and eliminated need for shimming tools particularly
carry the potential to widen access to this modality. However, a careful
balance between these benefits and maintenance of adequate image quality given
the reduced signal-to-noise ratio (SNR) is required and often leads to a
choice of thicker slices to increase the SNR. Thicker slices, however, put
further emphasis on accurate planning.

Recent work highlighted the use of AI methods during the acquisition in motion
detection, correction and automatic planning. Specifically in fetal MRI, work
showed the ability to perform quality control [14], automatic segmentation
[15, 16], and automatic tracking [17, 18]. Automatic field-of-view
prescription was shown in the abdomen using deep learning segmentations [21]
and in the heart using tracking based on landmarks [19] with successful
detection ratings of 99.7%-100% for cine images and Euclidean distances
between manual and automatically detected labels from 2 to 3.5 mm. Similar
work in the brain detecting landmarks such as the anterior and posterior
commissures and subsequently the symmetry line using multi-task deep neural
networks were demonstrated among others by Yang et al [20]. Specifically for
fetal MRI, segmentation of the eye region and detection of the general head
position was suggested by Hoffmann et al [22] using classical image processing
methods such as maximally stable extremal regions and by Xu et al [23] using
convolutional networks - both applicable for real-time slice planning in the
future.

Here, we present an automatic, fast landmark detection and subsequent
automatic radiological planning of fetal brain scans. The technique was
implemented, tested, and evaluated on prospective 0.55T low-field fetal MRI
scans.

## Methods

A method allowing automatic planning of the three radiological fetal brain
planes using landmarks is presented and compared to manual planning. The
methods section will first detail the conventional manual planning and then
describe the proposed new method.

#### Manual planning of brain acquisitions

Planning radiological fetal brain planes typically requires whole uterus ssTSE
acquisitions as a starting point. An example is given in Figure 1. The orbits
of the eyes and the back of the skull (Figure 1A, blue arrows) are used to
define the orientation of the true brain axial scan, with final adjustments
accomplished using the bottom of the lobes (Figure 1B, blue arrows, dotted
blue line), slightly shifting the centre of the acquisition towards the centre
of the brain (Figure 1B, blue line showing through-plane view). The midline
between the two hemispheres allows to guide the true brain sagittal
orientation (Figure 1B, dotted pink line, through-plane view). A favourable
view of the lobes is shown in Figure 1C, which is subsequently used to define
the true brain coronal acquisition (blue box, in-plane view). After the true
brain axial scan is acquired, final adjustments to the coronal and sagittal
radiological planes are performed, for instance by checking that the sagittal
centre of acquisition matches the brain midline across multiple axial slices
(Figure 1D-F, pink lines showing sagittal through-plane view). The dotted
lines show preliminary planning and the full lines show the final planning of
the scan.

Figure 1: Conventional manual planning of the three radiological orientations
for fetal anatomical T2-weighted single-shot Turbo Spin Echo (ssTSE)
acquisitions. A-C illustrate whole uterus ssTSE slices in sagittal and coronal
orientation to the uterus overlaid with the planning lines (cyan) to obtain
(D) an axial brain ssTSE stack, allowing together with the uterus stacks to
plan true sagittal (pink planning lines) and coronal (dark blue planning
lines) brain ssTSE stacks. The blue arrows in A-C illustrate the used
landmarks during manual planning.

#### Automatic planning of brain acquisitions

The automatic sequence planning framework was implemented on a 0.55T scanner
(MAGNETOM Free.Max, Siemens Healthcare, Erlangen, Germany). A whole-uterus
multi-echo gradient-echo single-shot echo planar imaging (EPI) sequence in
coronal maternal orientation is acquired in less then 30 seconds to perform
T2* mapping in several fetal organs and the placenta. In this study, this
sequence is used in addition for the automatic detection of fetal brain
structures enabling the automatic planning. The proposed solution consists of
three steps: (A) The fetal brain position is detected on the EPI scan and a
bounding box is generated, (B) The landmarks in the fetal head are identified,
(C) The true fetal brain planes are calculated and applied to the following
high-resolution ssTSE sequences (see Figure 2 for an overview of the process
and Table 1 for all used imaging parameters).

Figure 2: Schematic overview over the entire pipeline for automatic planning. (A) Detection of the fetal brain landmarks in the EPI sequence and (B) Planning parameters applied in the ssTSE sequence. Dataset | Parameters | Subjects | Time
---|---|---|---
EPI training 1.5T | 1.5T Philips Ingenia, 28-channel torso coil |  |
| Matrix=144x144-288x288, Resolution 2.5 mm isotropic, 30–96 slices |  |
| TE=[14.6/77.4/140.1/202.8/265.5] ms | 80 | 22.5 s
EPI training 3T | 3T Philips Achieva, 32-channel cardiac coil |  |
| Matrix=144x144-192x192, Resolution 2/3 mm isotropic, 45–75 slices |  |
| TE=[3.8/70.4/127/183.6] / [10.1/54.3/98.4/142.5/186.8] ms | 77 | 18.3 s
EPI autoplan | 0.55T Siemens MAGNETOM Free.Max, 6-channel coil, 9-channel spine coil |  |
| Matrix=100x100-128x128, Resolution 3.13–4.0 mm isotropic, |  |
| TE=81 ms, 50–59 slices | 9 | 25 s
ssTSE autoplan | 0.55T Siemens MAGNETOM Free.Max, 6-channel coil, 9-channel spine coil |  |
| Matrix=304x304, Resolution 1.48x1.48x4.5 mm3, 35 slices |  |
| TE=106 ms, TR=14.6 s | 9 | 55 s
Table 1: Sequence parameters for all described sequences used for training
(first two rows, white) and for the prospective automatic planning method
(last two rows, gray).

### (A) Localization of the fetal brain

First, the EPI sequence was modified to export the acquired raw data to a
Gadgetron reconstruction pipeline [24], deployed on an external GPU-equipped
(NVIDIA GEFORCE RTX 2080 Ti, NVIDIA Corporate, Santa Clara, CA) computer
connected to the internal network of the MRI scanner. The raw data is
converted to ISMRMRD format immediately upon acquisition and reconstructed
using off-the-shelf Gadgets that provide generic modules for configuring the
streaming reconstruction in the Gadgetron framework. Then, a Python Gadget was
implemented to automatically estimate the position of the region of interest
in the image using a pre-trained 3D UNet [25] for fetal brain localization. A
bounding box encompassing the fetal brain is calculated and used for the
following landmark detection task [18].

##### Datasets and training

The fetal brain localization network was trained on 125 labelled fetal EPI
datasets acquired at 1.5T/3T and tested on 29 0.55T fetal datasets. To
increase the robustness of the network, the EPI scans used for the training
and performance evaluation of the model deliberately vary in the acquisition
parameters (field strength, echo time, resolution, acceleration factor),
gestational age (15-40 weeks), fetal health (control cases, fetal growth
restriction, prolonged preterm rupture of the membranes, etc.), and fetal
position (cephalic, breech, transverse). The parameters for the training data
set were: 1) 3T Philips Achieva, 32-channel cardiac coil and 16-channel spine
coil, matrix size=144x144-192x192, isotropic resolution 2mm3,
TE=[3.8/70.4/127/183.6]/ [10.1/54.3/98.4/142.5/186.8] ms, 45–75 slices; 2)
1.5T Philips Ingenia, 28-channel torso coil, matrix size=144x144-288x288,
isotropic resolution2.5mm, TE=[14.6/77.4/ 140.1/202.8/265.5] ms, 30–96 slices.
The trained model was tested on low-field fetal datasets acquired on a 0.55T
Siemens MAGNETOM Free.Max, using a blanket-like BioMatrix Contour-L 6-channel
coil and fixed 9-channel spine coil, matrix size=100x100-128x128, isotropic
resolution 3.13–4.0 mm, TE=[46/120/194/268/342] ms, 50–59 slices.

### (B) Landmark detection

A deep learning-based landmark detection method using the nnUNet framework
[26] was adopted to extract specific head landmarks, concretely the orbit of
both fetal eyes and the lower edge of the cerebellum. This framework performs
semantic segmentation and it automatically adapts to a given dataset by
analysing the provided training data and configuring a matching UNet-based
segmentation pipeline. Furthermore, the furthest anterior point (FAP) of the
fetal brain mask is extracted. The centre of mass (CoM) coordinates of the
landmarks, the brain mask, and the FAP are then transformed into the patient
coordinate system and written into a text file on the external server and
scanner host. Figure 3A illustrates the key points extracted and used for
planning the radiological acquisitions.

Figure 3: A) Key points extracted from the whole-uterus coronal EPI scan for
automatic planning of the ssTSE scans are illustrated: these include key
landmarks eyes (red and green dots) and cerebellum (blue dot) and two points
extracted from the fetal brain: the brain centre of mass (CoM, yellow dot) and
the furthest anterior point. B) The axial orientation is defined using the
eyes and cerebellum centre of mass coordinates, the sagittal orientation uses
the eyes CoM coordinates, and the coronal orientation is defined using the
eyes CoM and furthest anterior point of the fetal brain mask. The brain CoM is
used to define the centre of the slice stack for all radiological planes. The
purple boxes illustrate the slice position for each orientation: in axial and
coronal planes the views shown are in-plane, and for the sagittal plane the
view shown is through-plane.

##### Datasets and training

The landmark nnUNet was trained and tested on cropped and labelled images of
76/15 fetal subjects, respectively. Automatic cropping was performed by first
applying the inference of the fetal brain localization network to the datasets
and using the estimated fetal brain position to define a bounding box that is
defined by the brain mask and an expansion factor of 50%, defining how much
the bounding box should be enlarged relative to the dimensions of the fetal
brain, ensuring that the cropped region contains all key landmarks. The gold
standard segmentations of the landmarks were manually drawn for each dataset.
All slice stacks were acquired using the low-field fetal protocol described
above.

### (C) Calculation of the radiological brain planes

#### Automatic orientation calculation of brain acquisitions

The ssTSE sequence that follows the EPI sequence in the protocol was modified
to use the information stored in the file to calculate the three standard
radiological planes axial, sagittal, and coronal. The normal vector to the
sagittal plane is calculated as the vector between the eyes landmarks (see
Figure 3B). It was identified that the eyes and FAP lie on the same
approximately coronal plane and the eyes and cerebellum lie on the same
approximately axial plane. Therefore, the normal vector to the coronal plane
was calculated as the dot product of the two vectors between each eye and the
FAP, and the normal vector to the axial plane as the dot product of two
vectors between each eye and the cerebellum. A hierarchical approach is chosen
to increase robustness even more: If the cerebellum is not extracted
successfully, the CoM of the brain is used instead for the vector calculation.
The CoM of the brain is used to define the centre of the slice stack for all
planes.

### Experiments and evaluation

#### Evaluation of the fetal brain localization and landmark detection
accuracy

The performance of the localization network was analyzed using the Dice
similarity coefficient (DSC) and the Intersection-over-Union (IoU) metric
calculated between the gold standard manual segmentations performed by two
fetal experts (9 years and 2 years of fetal MRI experience respectively) and
the brain masks generated by the network. The landmark detection performance
was analyzed by calculating the 3D distance between the centre of mass of the
manual segmentations of the landmarks performed by fetal MRI experts and the
landmarks obtained from the network.

#### Real-time fetal brain plane planning

The entire pipeline was acquired prospectively in nine pregnant volunteers in
St Thomas’ Hospital, recruited between October-December 2023 after informed
consent was obtained as part of two ethically approved studies (MEERKAT
REC19/LO/0852 and miBirth 23/LO/0685). Women were scanned on the above-
described clinical 0.55T MAGNETOM Free.Max scanner in the supine position with
leg support to ensure comfort with life monitoring throughout the scan.
Gestational ages ranged between 20.3 and 37.3 weeks (mean 32.8+-3.5). For each
fetal subject, the initial whole-uterus coronal multi-echo gradient-echo
single-shot EPI scan was acquired (resolution=3.13-4.0 mm isotropic,
TE=[81/227/372] ms, 50–59 slices, matrix size=128x128). Subsequently, the
automatically-planned ssTSE (autoplan-ssTSE) prototype sequence was acquired
in all three radiological planes with the following parameters:
resolution=1.48x1.48x4.5 mm3, TE=106 ms, 35 slices, matrix size=304x304,
TE=106 ms, TR=14.6 s. In addition, the same sequence was acquired with manual
planning of the three radiological planes by fetal radiographers (1-5 years of
experience) for all subjects. In total, for each fetal subject, three planes
(axial, coronal, sagittal) were acquired, resulting in a total of 27
acquisitions of each planning approach - manual and automatic. The time
required for planning by the radiographers and the automatic method was
measured.

#### Evaluation of the planning quality and diagnostic quality of the
prospectively acquired images

Quantitative evaluation was performed blinded to the method and the ability to
produce accurate measures was scored. The resulting automatically planned
radiological planes were assessed by a fetal radiographer with five years of
fetal MRI experience answering the questions "Rate the planning quality [1-3]"
(1 - re-acquisition required, 2 - usable, 3 - full brain coverage, symmetry of
the brain structures, no re-acquisition required). A fetal radiologist with
$>$ 15 years of experience additionally quantified the clinical value of the
automatic planning method by attempting to perform, for both manually and
automatically planned acquisitions, five measurements in the sagittal view
(corpus callosum, cerebellar vermis, pons anterior-posterior (AP) diameter,
pituitary stalk, hard palate), four in the coronal view (trans-cranial
diameter (TCD), bi-parietal diameter (BPD), cavum septum pellucidum (CSP),
ventricle) and five in the axial view (BPD, occipito-frontal diameter (OFD),
ventricle, TCD, CSP). The radiologist was asked to answer "Can you perform all
radiological measurements? [0-5]" (0 - unusable, 5 - all five measurements
successfully performed). The results were in addition assessed against
gestational age and maternal Body Mass Index.

## Results

### (A) Fetal brain localization

The brain localization task, trained on mid/high-field data, achieved an
overall DSC of 0.82±0.18 and IoU of 0.73±0.19 when tested on low-field scans
across all TEs, fetal positions, and gestational ages. Extraction of fetal
brain masks took between 11.43 and 20.85 ms per volume in the offline testing
mode.

### (B) Landmark detection

The mean distance between the CoM of the eyes and cerebellum labels from
automatic and manual segmentations was 4.21$\pm$2.56 mm and 6.47$\pm$3.23 mm,
respectively. Figure 4 shows the predicted brain and landmarks segmentations
for one fetal subject of the landmarks model test set. Figure 5 shows examples
of predicted landmarks generated by the nnUNet, the corresponding ground-truth
segmentations, and the distance between the two. Landmark detection took 5.18
seconds per volume in the offline testing mode.

Figure 4: Two-step localisation task - global localisation of the fetal head
using a fetal brain segmentation 3D UNet model, and local localisation with
extraction of the 3 fetal head landmarks - eyes and cerebellum. Figure 5:
Ground-truth and predicted landmarks, generated by the nnUNet, for fetal
subjects 0, 4, 7, 8, 12, and 13 of the landmark detection test set. The
distance between the centre of mass of each is additionally displayed. The
model outputs the eyes landmarks using a single label, hence they are depicted
with the same colour (and label). This result is then processed to separate
the two unconnected regions to be used to guide the planning of the ssTSE
scan.

### (C) Real-time fetal brain plane planning

The entire pipeline was successfully run in all nine fetal scans, with all
steps of the approach successfully implemented and tested prospectively. The
estimated time between the completion of the EPI sequence and the start of the
ssTSE sequence, ready to be run with the planning parameters correctly set, is
less than 5 seconds with the automatic method, and an average of 1.5 minutes
for the specialist radiographer and 4 minutes for the novice radiographer.
Landmark detection and automatic radiological planning are shown in Figure 6
alongside the radiographer’s manually planned acquisitions, with matching
anatomical structures in the manual and automatic acquisitions highlighted to
demonstrate that both techniques allow the same structures of interest to be
captured comparably. The anatomies (green squares, blue arrows) are labelled
between 1-10, with the corresponding dictionary at the bottom.

Figure 6: Landmarks extracted from whole uterus coronal EPI scans and
radiographer-planned and automatic (orientation defined from the position of
the eyes and cerebellum, extracted from the EPI) radiological ssTSE
acquisitions. Anatomical structures of interest were highlighted (green
squares, blue arrows) for manual and automatic acquisitions to demonstrate
comparable orientation planning.

In Figure 7A, a quantitative assessment of the real-time landmark detection
task applied to the prospective whole uterus EPI scans is depicted. The
landmarks were extracted in real-time during the scan to automatically plan
the radiological brain acquisitions and were subsequently segmented manually
by two observers M1 and M2, where M2 is the observer that segmented the
landmarks for training the nnUNet. All landmarks but one cerebellum
segmentation (fetal subject 9 of GA 20 weeks) were prospectively extracted
during the scan. The inability of the model to extract the landmark may be
linked to the early gestation of the fetus and the consequent small size of
the brain structures, as well as the fact that early gestation fetuses are not
as well represented in the training dataset of the model, a natural bias that
is often present in fetal MRI studies. For this case, the CoM of the brain was
used instead as a landmark to allow for the automatic planning to still be
performed. For the 9 fetal subjects scanned, the differences in distances (in
mm) between the CoM of eyes and cerebellum landmarks segmented by M1, M2, and
automated (A) were calculated. For the eyes, the difference between M1 and A
ranged between 1.25-21.81 mm, M2-A between 1.47-10.13 mm, and M1-M2 between
1.54-11.98 mm. As for the cerebellum, the CoM differences in distance for M1-A
ranged between 9.43-41.88 mm, M2-A between 1.99-12.1 mm, and M1-M2 between
7.11-38.91 mm. While inter-observer variability never exceeded 11.98 mm for
the eyes landmarks, the cerebellum presented much larger variability in the
segmentations produced by M1 and M2, which can be explained by the boundaries
of this anatomical structure not being as well defined as for the eyes,
leading to variability in the size of the segmentation. For this reason, using
a deep learning approach helps to eliminate observer bias, as segmentations
are more standardised and consistent.

The quantitative evaluation of the three achieved radiological planes is
displayed in Figure 7B and C and Figure 8. In both Figures, the assessments
performed by the radiographer and radiologist aimed to score how close to a
perfect true brain plane each acquisition was and the clinical value of the
automatically planned images (compared with the manually planned ones). With 3
radiological planes (axial, coronal, sagittal) acquired for each fetal
subject, 27 planes were acquired in total in this work. Out of 27
automatically planned acquisitions, 22 scans were successfully acquired
without the need for re-acquisition, and 24 out of 27 manually planned scans
similarly did not require repeating according to the analysis performed by a
fetal radiographer. Full brain coverage was achieved in all radiological
planes for all manual and automatic scans, thus the need for re-acquisition
resided in large tilts present in the orientation of the brain, with
asymmetries visible between the two brain hemispheres and anatomical
structures of interest present in both.

In 27 manually planned scans, 6 scans displayed slight asymmetry with small
rotations of the anatomy that would not affect the clinical value of the
images and thus did not require re-acquisition (4 coronal planes and 2 axial).
Severe asymmetry was observed in 3 acquisitions (2 axial planes and 1
sagittal) with re-acquisition needed. The remaining 18 acquisitions showed a
complete symmetry of the two brain hemispheres. Regarding the automatic
planning method, full symmetry of the brain hemispheres and neurological
structures was achieved in 17 automatic acquisitions, demonstrating the
ability of the method to plan true brain planes accurately. Generating the
automatic acquisition plane took, on average, 5 seconds, and the radiological
acquisitions were acquired in less than one minute each. Asymmetry was
observed in 10 cases, although at different degrees: 5 scans showed larger
asymmetry with severe tilts (3 axial planes, 1 coronal, and 1 sagittal), and 5
showed smaller tilts causing slight asymmetry which would not affect the
clinical value of acquisitions and therefore would not require re-acquisition
(2 coronal planes, 2 sagittal, and 1 axial). Figure 7B illustrates the
radiographer assessments. In summary, a quality score of 81.5% (2.44/3) was
achieved by the automatic planning method and 85.2% (2.56/3) for manual
planning.

The clinical value and image quality were additionally evaluated by a fetal
radiologist. With a maximum score of 14 measurements per fetal subject, the
number of measurements successfully performed in the manually planned
acquisitions was: 14, 10, 7, 6 (2 subjects), 5 (2 subjects, 1 of GA 20 weeks),
2, and 1. Artefacts were also visible in 3 coronal acquisitions, which may
have affected the ability to perform measurements. Using the automatic
planning method, the number of measurements performed was: 12, 11 (2
subjects), 7 (one plane showing artefacts), 5 (2 subjects), 4 (fetal subject
of GA 20 weeks), 3, and 0 (one plane showing distortion). An average of 2.07/5
measurements were successfully performed by a radiologist for the manual
acquisitions, and 2.14/5 for the automatic acquisitions.

The measurements with the highest scores across all fetal subjects and all
orientations were the CSP and the ventricle, in the coronal orientation, with
scores of manual 8/9 and automatic 6/9 subjects for both measurements. The
corpus callosum (manual 1/9, automatic 0/9 subjects) and pons AP diameter
(manual 1/9, automatic 2/9 subjects), both in sagittal orientation, showed the
lowest scores. For both sagittal and coronal orientations, radiographer-
planned acquisitions scored slightly higher than the automated ones (by 2.22%
and 5.0%, respectively), however, the automatically-planned axial orientation
was superior to the manual planning by 11.1%. Overall, the automatic method
for planning the acquisitions was 1.59% higher than the manual planning
approach, according to the image quality quantitative assessment performed by
a radiologist.

As illustrated in Figure 7C, out of 27 manual scans 5 measurements were
performed in 2 scans, 4 measurements in 4 scans, 3 in 4 scans, 2 in 5 scans, 1
in 3 scans, and 1 dataset severely affected by motion, resulting in 8 scans
without immediate clinical value (unless tools for rotating the brain anatomy
are utilised). Regarding the automatically-planned scans, 5 measurements were
performed in 4 scans, 4 measurements in 6 scans, 3 in 5 scans, and 2 in 2
scans. One dataset was similarly corrupted by motion, resulting in 9 scans
without immediate clinical value.

Figure 7: Quantitative evaluation of the 27 prospectively acquired ssTSE data
sets. (A) Landmark assessment: Difference in mm between the manual
segmentations performed by two observers (M1 and M2) and the automatic method
(A) for the eyes and the cerebellum. (B) Radiographer assessment: Image
quality for all 27 acquisitions for the manual and automatic acquisitions. (C)
Radiologist assessment: Image quality for all 27 acquisitions for the manual
and automatic acquisitions.

Figure 8 depicts planning quality, assessed by a radiographer, and image
quality, assessed by a radiologist, according to gestational age (GA at scan)
and maternal Body Mass Index (BMI).

Figure 8: Quantitative assessment of the planning quality (A-B) and image
quality (C-D) illustrated for all three planes (coronal, axial, and sagittal)
against GA (A, C) and maternal BMI (B, D).

## Discussion and Conclusion

In this study, a fully automated framework for planning all three radiological
fetal brain planes was demonstrated in low-field MRI scans. The method
presented uses a whole-uterus coronal multi-echo EPI sequence, acquired and
employed for T2* relaxometry, to serve the additional purpose of fetal head
landmark detection. The brain and key landmarks are automatically extracted
and handed to the subsequent ssTSE acquisition to plan fetal brain-oriented
acquisitions in under 5 seconds. This automatic planning of fetal MRI scans,
shown here as comparable to the manual planning for clinical reporting,
reduces dependence on specialist staff and increases time efficiency and thus
carries the potential to significantly widen accessibility to fetal MRI beyond
specialist centres.

In line with recent work [22], the eyes were successfully used as landmarks
and were crucial for calculating the orientation of the fetal brain, however,
the detection shown here did not include further pose estimation or image
processing steps. While recent work employing AI to detect landmarks and
organs on the MR acquisitions focused on 1.5T and 3T [27, 18, 22, 19, 23, 14],
the present work was performed on low field fetal MRI at 0.55T, providing
increasing challenges regarding SNR and resolution, but allowing to take key
steps towards wider accessibility of fetal MRI.

The retrospective evaluation of the network that extracts the head landmarks
(eyes and cerebellum), resulted in an average distance between the CoM of the
manually segmented and automatically extracted eyes of 4.21 mm, which
corresponds to 1.34 voxels, and 6.47 mm for the cerebellum, corresponding to
2.07 voxels, which is sufficiently robust for the calculation of the
radiological planes. Fetal subjects 8 and 13 of the test set showed the
largest distances (mm) between the CoM of manual and automatic landmark masks,
however, Figure 5 demonstrates how even at such large distances (9.44 mm and
16.84 mm for eyes and cerebellum of Fetus 13) the predictions are still highly
comparable to the gold standard segmentations and extract the key landmarks
accurately enough for the orientation calculations of the ssTSE scans.
According to the quantitative and qualitative assessments of the calculation
of the radiological brain planes, the automatically planned scans achieved
comparable image quality to the scans manually planned by an experienced fetal
radiographer.

Major strengths of the current study are the complete and robust real-time
deployment. The two-step approach presented here for landmark detection: first
the identification of the brain with a 3D UNet followed by the landmark
detection using nnUNet and the hierarchical application of landmarks in cases
where individual landmarks are not identified - e.g. replacing the cerebellum
by the centre of the brain for the sagittal prescription - contributes to
robustness even in younger fetuses. Furthermore, the developed method was
built using open-source frameworks, facilitating straightforward translation
and dissemination. All steps, including the EPI reconstruction with off-the-
shelf Gadgets, the fetal brain localization and landmark detection trained
models, and Gadget are available to any interested researchers
(https://github.com/saranevessilva/fetal-brain-landmarks). All anonymized
fetal data is also available upon request.

There are, however, limitations with the current study. First, the present
study is a single-center study. The training data and retrospectively assessed
test data are small, as well as the number of prospective cases, which
currently does not allow a more robust evaluation of the method in a larger
cohort. This is often the case in fetal MRI as such a modality is not yet
widely available worldwide and data sharing policies between institutions and
studies are often restrictive. The framework may need to be further refined to
perform robustly in wider populations and across different scanners/field
strengths. Future work will hence investigate further developments to ensure
the robustness of the automatic planning framework in this wider context. EPI
images are reconstructed using standard Fourier reconstruction computed with
off-the-shelf Gadgets (without image filtering or AI-based image enhancement
provided by the scanner reconstruction pipeline) resulting in limited SNR,
which may hinder the performance of the landmark detection. This will be
addressed by further improving the real-time reconstruction pipeline, by
including image filtering or AI-based image enhancement to achieve increased
SNR in the EPI images. Fetal MRI at higher field will suffer less from this
limitation. Next, this framework is based on a whole-uterus multi-echo EPI
sequence, not requiring any planning and fitting to the standard patient
examination as current practice for all fetal MRI scans in our institution.
However, the reduced resolution of this EPI sequence (3 mm isotropic) compared
to the ssTSE sequence (1.25 mm in-plane) might influence the achieved
precision regarding landmark detection. The network and achieved complete
automatic planning can however be extended to whole uterus ssTSE localiser
scans in the future. Furthermore, while the landmarks were chosen as
independently as possible from brain structures involved in common pathologies
such as the ventricles and the corpus callosum - and in line with recent work
[22] \- the automatic planning in fetuses with developmental abnormalities in
the location of the lower edge of the cerebellum or the orbits of the eyes
might involve the need for additional manual adjustment. Finally, adding a
fourth landmark in the back of the skull might be helpful to further stabilize
the planning of the acquisition in the axial orientation.

Future work towards an automatic fetal brain MR examination will furthermore
include the real-time deployment of automatic quality control [14], sub-
regional segmentation [10] and automatic biometry [28].

To conclude, this study shows the feasibility of rapid deep learning-based
automatic planning of anatomical radiological fetal brain scans and presents
an open-source framework open for further extensions and improvements. The
clear next application is fetal cardiac landmark detection for automatic
planning of phase contrast sequences acquired to study the flow in the major
arteries.

## Acknowledgements

The authors thank all pregnant women and their families for taking part in
this study, the midwives Imogen Desforges, Chidinma Iheanetu Oguejiofor, and
Maggie Lee, and clinical research fellow Simi Bansal and Vanessa Kyriakopoulou
for their invaluable efforts in recruiting and looking after the women in this
study as well as Massimo Marenzana and Michela Cleri for their involvement in
the acquisition of these datasets. This work was supported by a Wellcome Trust
Collaboration in Science grant [WT201526/Z/16/Z], Heisenberg funding from the
DFG [502024488], a UKRI FL fellowship, an NIHR Advanced Fellowship to LS
[NIHR3016640], an EPSRC Research Council DTP grant [EP/R513064/1] and by core
funding from the Wellcome/EPSRC Centre for Medical Engineering
[WT203148/Z/16/Z]. The views presented in this study represent those of the
authors and not of Guy’s and St Thomas’ NHS Foundation Trust.

## References

  * [1] Alford, R.E., Bailey, A.A., Twickler, D.M. (2016).  Fetal Central Nervous System.  In: Masselli, G. (eds) MRI of Fetal and Maternal Diseases in Pregnancy.  Springer, Cham.  pp. 91–118.  https://doi.org/10.1007/978-3-319-21428-3_6.
  * [2] Prayer, D., Brugger, P.C., Nemec, U., Milos, R.I., Mitter, C., Kasprian, G. (2010).  Cerebral Malformations.  In: Prayer, D. (eds) Fetal MRI.  Medical Radiology, pp. 287–308.  Springer, Berlin, Heidelberg.  https://doi.org/10.1007/174_2010_117.
  * [3] Epstein, K.N., Kline-Fath, B.M., Zhang, B., Venkatesan, C., Habli, M., Dowd, D., Nagaraj, U.D.  Prenatal Evaluation of Intracranial Hemorrhage on Fetal MRI: A Retrospective Review.  American Journal of Neuroradiology, 42(12):2222–2228, 2021.  DOI: 10.3174/ajnr.A7320.  https://www.ajnr.org/content/42/12/2222.
  * [4] van der Hoek-Snieders, H. E. M., van den Heuvel, A. J. M. L., van Os-Medendorp, H., Kamalski, D. M. A.  Diagnostic accuracy of fetal MRI to detect cleft palate: a meta-analysis.  Eur. J. Pediatr., 179(1):29–38, January 2020.  https://link.springer.com/article/10.1007/s00431-019-03526-7.
  * [5] Cornejo, P., Feygin, T., Vaughn, J., Pfeifer, C. M., Korostyshevska, A., Patel, M., et al.  Imaging of fetal brain tumors.  Pediatr. Radiol., 50(13):1959–1973, December 2020.  https://link.springer.com/article/10.1007/s00247-020-04664-1.
  * [6] Kyriakopoulou, V., Vatansever, D., Davidson, A., Patkee, P., Elkommos, S., Chew, A., et al.  Normative biometry of the fetal brain using magnetic resonance imaging.  Brain Structure & Function, 222(5):2295–2307, July 2017.  DOI: 10.1007/s00429-016-1345-8.  https://link.springer.com/article/10.1007/s00429-016-1345-8.
  * [7] Gholipour, A., Rollins, C. K., Velasco-Annis, C., Ouaalam, A., Akhondi-Asl, A., Afacan, O., et al.  A normative spatiotemporal MRI atlas of the fetal brain for automatic segmentation and analysis of early brain growth.  Sci Rep, 7(476), 2017.  https://www.nature.com/articles/s41598-017-00516-6.
  * [8] Uus, A., Zhang, T., Jackson. L. H., Roberts, T. A., Rutherford, M. A., Hajnal, J. V., Deprez, M.  Deformable Slice-to-Volume Registration for Motion Correction of Fetal Body and Placenta MRI.  IEEE Trans Med Imaging, 39(9):2750–2759, September 2020.  DOI: 10.1109/TMI.2020.2972867.  https://ieeexplore.ieee.org/document/8971472.
  * [9] Kuklisova-Murgasova, M., Quaghebeur, G., Rutherford, M. A., Hajnal, J. V., Schnabel, J. A.  Reconstruction of fetal brain MRI with intensity matching and complete outlier removal.  Med Image Anal, 16(8):1550–1564, December 2012.  DOI: 10.1016/j.media.2012.06.001.  https://www.sciencedirect.com/science/article/pii/S1361841512000845.
  * [10] Uus, A. U., Kyriakopoulou, V., Makropoulos, A., Fukami-Gartner, A., Cromb, D., Davidson, A., et al.  BOUNTI: Brain vOlumetry and aUtomated parcellatioN for 3D feTal MRI.  bioRxiv, 2023.  DOI: 10.1101/2023.04.18.537347.  https://www.biorxiv.org/content/early/2023/04/27/2023.04.18.537347.
  * [11] Aviles Verdera, J., Story, L., Hall, M., Finck, T., Egloff, A., Seed, P. T., et al.  Reliability and Feasibility of Low-Field-Strength Fetal MRI at 0.55 T during Pregnancy.  Radiology, 309(1):e223050, 2023.  DOI: 10.1148/radiol.223050.  https://doi.org/10.1148/radiol.223050.
  * [12] Ponrartana, S., Nguyen, H. N., Cui, S. X., Tian, Y., Kumar, P., Wood, J. C., Nayak, K. S.  Low-field 0.55 T MRI evaluation of the fetus.  Pediatr. Radiol., March 2023.  DOI: 10.1007/s00247-023-05151-8.  https://link.springer.com/article/10.1007/s00247-023-05151-8.
  * [13] Payette, K., Uus, A., Aviles Verdera, J., Avena Zampieri, C., Hall, M., Story, L., et al.  An automated pipeline for quantitative T2* fetal body MRI and segmentation at low field.  ArXiv, August 2023.  https://arxiv.org/abs/2308.02132.
  * [14] Gagoski, B., Xu, J., Wighton. P., Tisdall. M. D., Frost, R., Lo, W. C., Golland, A. van der Kouwe, E. Adalsteinsson, P. E. Grant.  Automated detection and reacquisition of motion-degraded images in fetal HASTE imaging at 3 T.  Magn Reson Med, 87(4):1914–1922, April 2022.  DOI: 10.1002/mrm.29095.  https://onlinelibrary.wiley.com/doi/full/10.1002/mrm.29095.
  * [15] Faghihpirayesh, R., Karimi, D., Erdogmus, D., Gholipour, A.  Deep Learning Framework for Real-time Fetal Brain Segmentation in MRI.  arXiv, May 2022.  https://arxiv.org/abs/2205.01675.
  * [16] Salehi, S. S. M., Hashemi, S. R., Velasco-Annis, C., Ouaalam, A., Estroff, J. A., Erdogmus, D., et al.  Real-time automatic fetal brain extraction in fetal MRI by deep learning.  In: 2018 IEEE 15th International Symposium on Biomedical Imaging (ISBI 2018), pp. 720–724, April 2018.  DOI: 10.1109/ISBI.2018.8363542.  https://ieeexplore.ieee.org/document/8363542.
  * [17] Singh, A., Salehi, S. S. M., Gholipour, A.  Deep Predictive Motion Tracking in Magnetic Resonance Imaging: Application to Fetal Imaging.  IEEE Transactions on Medical Imaging, 39(11):3523–3534, 2020.  DOI: 10.1109/TMI.
  * [18] Neves Silva, S., Aviles Verdera, J., Tomi-Tricot, R., Neji, R., Uus, A., Grigorescu, I., et al.  Real-time fetal brain tracking for functional fetal MRI,  Magn. Reson. Med.,  Jul 2023,
  * [19] Xue, H., Artico, J., Fontana, M., Moon, J. C., Davies, R. H., Kellman, P.  Landmark Detection in Cardiac MRI by Using a Convolutional Neural Network,  Radiol Artif Intell,  Sep 2021,
  * [20] Yang, X., Tang, W. T., Tjio, G., Yeo, S. Y., Su, Y.  Automatic detection of anatomical landmarks in brain MR scanning using multi-task deep neural networks.  Neurocomputing, 396:514–521, July 2020.  DOI: 10.1016/j.neucom.2020.04.078.
  * [21] Lei, K., Syed. A. B., Zhu, X., Pauly, J. M., Vasanawala, S. V.  Automated MRI Field of View Prescription from Region of Interest Prediction by Intra-Stack Attention Neural Network.  Bioengineering (Basel), 10(1), January 2023.  DOI: 10.3390/bioengineering10010005.
  * [22] Hoffmann, M., Turk, E. A., Gagoski, B., Morgan, L., Wighton, P., Tisdall, M. D., et al.  Rapid head-pose detection for automated slice prescription of fetal-brain MRI.  Int. J. Imaging Syst. Technol., 31(3):1136–1154, September 2021.  DOI: 10.1002/ima.22651.
  * [23] Xu, J., Zhang, M., Turk, E. A., Zhang, L., Grant, P. E., Ying, K., et al.  Fetal Pose Estimation in Volumetric MRI Using a 3D Convolution Neural Network.  In: Medical Image Computing and Computer Assisted Intervention – MICCAI 2019, pp. 403–410, 2019.
  * [24] Hansen, M. S., Sørensen, T. S.  Gadgetron: an open source framework for medical image reconstruction.  Magn Reson Med, 69(6):1768–1776, June 2013.  DOI: 10.1002/mrm.24383.
  * [25] Uus, A., Grigorescu, I., van Poppel, M., Hughes, E., Steinweg, J., Roberts, T., et al.  3D UNet with GAN discriminator for robust localization of the fetal brain and trunk in MRI with partial coverage of the fetal body.  bioRxiv, 2021.  DOI: 10.1101/2021.06.23.449574.  https://www.biorxiv.org/content/early/2021/06/24/2021.06.23.449574.
  * [26] Isensee, F., Jaeger, P. F., Kohl, S. A. A., Petersen, J., Maier-Hein, K. H.  nnU-Net: a self-configuring method for deep learning-based biomedical image segmentation.  Nat. Methods, 18(2):203–211, February 2021.  DOI: 10.1038/s41592-020-01008-z.
  * [27] Salehi, S. S. M., Khan, S., Erdogmus, D., Gholipour, A.  Real-Time Deep Pose Estimation With Geodesic Loss for Image-to-Template Rigid Registration.  IEEE Transactions on Medical Imaging, 38(2):470–481, 2019.  DOI: 10.1109/TMI.2018.2866442.
  * [28] She, J., Huang, H., Ye, Z., Huang, W., Sun, Y., Liu, C., et al.  Automatic biometry of fetal brain MRIs using deep and machine learning techniques.  Sci. Rep., 13(1):17860, October 2023.  DOI: 10.1038/s41598-023-17202-0.


# On the structure of even $K$-groups of rings of algebraic integers

Meng Fai Lim111School of Mathematics and Statistics $\&$ Hubei Key Laboratory
of Mathematical Sciences, Central China Normal University, Wuhan, 430079,
P.R.China. E-mail<EMAIL_ADDRESS>
###### Abstract

In this paper, we describe the higher even $K$-groups of the ring of integers
of a number field in terms of class groups of an appropriate extension of the
number field in question. This is a natural extension of the previous
collective works of Browkin, Keune and Kolster, where they considered the case
of $K_{2}$. We then revisit the Kummer’s criterion of totally real fields as
generalized by Greenberg and Kida. In particular, we give an algebraic
$K$-theoretical formulation of the said criterion which we will prove using
the algebraic $K$-theoretical results developed here.

Keywords and Phrases: Even $K$-groups, class groups, Kummer’s criterion.

Mathematics Subject Classification 2020: 11R42, 11R70, 19F27.

##  1 Introduction

Throughout, $p$ will always denote a fixed odd prime. Let $F$ be a number
field with ring of integers $\mathcal{O}_{F}$. Inspired by the work of Tate
[30], Browkin, Keune and Kolster have made extensive studies in comparing the
$K_{2}$-group $K_{2}(\mathcal{O}_{F})/p^{n}$ with the $S$-class group
$\operatorname{Cl}_{S}(F(\mu_{p^{n}}))$ of $F(\mu_{p^{n}})$ in a series of
papers [4, 12, 14, 15]. We should perhaps mention that Coates [6] has also
performed a study of this sort but instead over an (infinite) cyclotomic
$\mathbb{Z}_{p}$-extension (also see the related works of Gras [9] and Jaulent
[11]).

In this paper, we are interested in extending the works of Browkin, Keune and
Kolster to the higher even $K$-groups of $\mathcal{O}_{F}$. In slightly more
detail, let $i$ be a positive integer. Write $a:=a(F)$ for the largest integer
such that $F(\mu_{p})=F(\mu_{p^{a}})$, and set $b=b(i)$ to be the largest
integer such that $p^{b}$ divides $i$. Our main theorems consist of comparing
the group $K_{2i}(\mathcal{O}_{F})/p^{n}$ with certain eigenspaces of the
$S$-class group of an appropriate cyclotomic extension of $F$. In particular,
if $n\leq a+b$, we compare $K_{2i}(\mathcal{O}_{F})/p^{n}$ with a certain
eigenspace of the $S$-class group of $F(\mu_{p})$ (see Theorem 4.2 for the
precise statement). When $n>a+b$, the comparison is done over the field
$F(\mu_{p^{n-b}})$ (see Theorem 4.5). Note that in the case of $K_{2}$, the
quantity $b$ is always zero and so the comparison for large $n$ is always done
over $F(\mu_{p^{n}})$. However, for a higher even $K$-group, the presence of
$b$ requires extra care during our discussion.

As illustrations of our results, we specialize them to the context of a
quadratic field, where the results become slightly more explicit. In the
context of $K_{2}$, such $p^{n}$-rank results have also been worked out by
Browkin [4] and Zhou [36]. Our discussion here thus generalizes these prior
results. We note that many authors have applied the results of Browkin and
Keune to obtain explicit results for $K_{2}$ in various specific number fields
(for instance, see [5, 17, 25, 26, 35, 37, 38, 39], where this list is far
from being exhaustive). It would definitely be an interesting project to apply
the results of this paper to perform analogous studies on the higher even
$K$-groups for these said number fields. In this note, we shall only contend
ourselves with the case of a quadratic field (although we hope to return to
this subject in future works).

We next turn our attention to the case of a totally real field, where we give
an algebraic $K$-theoretical formulation of the Kummer’s criterion in the
sense of Greenberg [10] and Kida [13] (see Theorem 6.1 and Remark 6.2 for the
details). We then prove this via the algebraic $K$-theoretical results
developed in this paper. In particular, our proof does not make use of the
$p$-adic $L$-function.

We end the introductory section giving an outline of the paper. In Section 2,
we collect certain basic algebraic notion and notation required for our
discussion. Section 3 is where we recall the relation between the Sylow
$p$-subgroups of the even $K$-groups and various Galois cohomology groups.
This paves the way for us to prove our main results in Section 4. Section 5 is
where we specialize our main results to the context of a quadratic field.
Finally, in Section 6, we give the algebraic $K$-theoretical formulation of
the Kummer’s criterion of a totally real field and its proof.

### Acknowledgement

The author would like to thank Georges Gras for his interest and comments on
the paper. He is also grateful to the referees for their many valuable
comments and suggestions. This research is supported by the National Natural
Science Foundation of China under Grant No. 11771164.

##  2 Algebraic preliminaries

As a start, we introduce certain terminology and notation that will be used
throughout. Let $p$ be a prime. For a positive integer $t$, write $v_{p}(t)$
for the $p$-adic valuation of $t$. In particular, if $t$ is a $p$-power, we
have $t=p^{v_{p}(t)}$. For a finite abelian group $N$, denote by $N[p^{n}]$
the subgroup of $N$ consisting of elements annihilated by $p^{n}$. Plainly,
one has $N[p^{n-1}]\subseteq N[p^{n}]$ for $n\geq 1$, where $N[p^{0}]$ is
understood to be the trivial group. The quotient module $N[p^{n}]/N[p^{n-1}]$
can be viewed as a $\mathbb{F}_{p}$-vector space, where $\mathbb{F}_{p}$ is
the finite field with $p$ elements. Therefore, it makes sense to define the
$p^{n}$-rank
$r_{p^{n}}(N)=\dim_{\mathbb{F}_{p}}\big{(}N[p^{n}]/N[p^{n-1}]\big{)}$.

Throughout the paper, we shall also write $N/p^{n}$ for $N/p^{n}N$. The
following elementary lemma is left to the reader as an exercise.

######  Lemma 2.1.

Let $N$ be a finite abelian group. Then we have

$r_{p^{n}}(N)=r_{p^{n}}(N/p^{n}).$

In this paper, we are interested in studying the quantity
$r_{p^{n}}\big{(}K_{2i}(\mathcal{O}_{F})\big{)}$ which, by the preceding
lemma, leads us to analyse
$r_{p^{n}}\big{(}K_{2i}(\mathcal{O}_{F})/p^{n}\big{)}$. We shall frequently
make use of this viewpoint in the paper without any further mention.

Finally, for a given profinite group $G$ and a $G$-module $M$, we let $M^{G}$
be the subgroup of $M$ consisting of elements fixed by $G$ and $M_{G}$ be the
largest quotient of $M$ on which $G$ acts trivially. If $M$ is a discrete
$G$-module, we write $H^{k}(G,M)$ for the $k$-th cohomology group of $G$ with
coefficients in $M$.

##  3 Arithmetic preliminaries

For the discussion of our main results, we need to recall the relation between
the algebraic $K$-groups and étale/Galois cohomology. This relation is
essentially well-known (for instances, see [24, 31, 33]), but for the
convenience of the readers, we shall collect some aspects of the said relation
in this section. Again, we emphasize that we will always work under the
standing assumption that the prime $p$ is odd.

###  3.1 $i$-fold tensor product

To begin with, fix once and for all an algebraic closure $\bar{\mathbb{Q}}$ of
$\mathbb{Q}$. Therefore, an algebraic (possibly infinite) extension of
$\mathbb{Q}$ will mean an subfield of our fixed algebraic closure
$\bar{\mathbb{Q}}$. In particular, a number field is understood to be a finite
extension of $\mathbb{Q}$ contained in $\bar{\mathbb{Q}}$.

Let $F$ be a number field, whose ring of integers is in turn denoted by
$\mathcal{O}_{F}$. Throughout the paper, $S$ will always denote the finite set
of primes of $F$ which consists precisely of the primes above $p$ and the
infinite primes. We then write $\mathcal{O}_{F,S}$ for the ring of
$S$-integers. Let $F_{S}$ be the maximal algebraic extension of $F$ unramified
outside $S$ and denote by $G_{S}(F)$ the Galois group
$\operatorname{Gal}(F_{S}/F)$. For a given finite extension $L$ of $F$
contained in $F_{S}$, we shall write $S(L)$ for the set of primes of $L$ above
$S$. Equivalently, $S(L)$ is the set of primes of $L$ which consists of the
primes above $p$ and the infinite primes. Since $L$ is contained in $F_{S}$,
it follows from a straightforward verification that $F_{S}=L_{S(L)}$. We then
write $G_{S}(L)$ for the Galois group $\operatorname{Gal}(F_{S}/L)$.

Now, a $p$-primary discrete $G_{S}(F)$-module $M$ may also be viewed as a
discrete $G_{S}(L)$-module via restriction of action. From which, we have
restriction map

$H^{k}(G_{S}(F),M)\longrightarrow H^{k}(G_{S}(L),M)$

and corestriction map

$H^{k}(G_{S}(L),M)\longrightarrow H^{k}(G_{S}(F),M)$

on cohomology. For the subsequent discussion of the paper, we require the
following standard facts which are recorded here for convenience.

######  Lemma 3.1.

Suppose that $M$ is a $p$-primary discrete $G_{S}(F)$-module. Let $L$ be a
Galois extension of $F$ contained in $F_{S}$. Then the following statements
are valid.

  1. (i)

The corestriction map induces an isomorphism

$H^{2}(G_{S}(L),M)_{\operatorname{Gal}(L/F)}\cong H^{2}(G_{S}(F),M)$

  2. (ii)

If the group $\operatorname{Gal}(L/F)$ is finite of order coprime to $p$, then
the restriction map induces an isomorphism

$H^{2}(G_{S}(F),M)\cong H^{2}(G_{S}(L),M)^{\operatorname{Gal}(L/F)}.$

###### Proof.

The first assertion follows from reading off the initial term of the Tate
spectral sequence (cf. [21, Theorem 2.5.3]). On the other hand, in view of
$\operatorname{Gal}(L/F)$ being finite with order coprime to $p$, the
Hochschild-Serre spectral sequence

$H^{r}(\operatorname{Gal}(L/F),H^{s}(G_{S}(L),M))\Longrightarrow
H^{r+s}(G_{S}(F),M)$

is concentrated on the line $r=0$ yielding

$H^{s}(G_{S}(F),M)\cong H^{s}(G_{S}(L),M)^{\operatorname{Gal}(L/F)}$

for every $s\geq 0$. In particular, taking $s=2$, we have the second
assertion. ∎

We now introduce the discrete modules that will be frequently considered in
this paper. Denoting by $\mu_{p^{n}}$ the cyclic group generated by a
primitive $p^{n}$th-root of unity, we then write $\mu_{p^{\infty}}$ for the
direct limit of the groups $\mu_{p^{n}}$. These have natural $G_{S}(F)$-module
structures. Furthermore, for an integer $i\geq 2$, the $i$-fold tensor
products $\mu_{p^{n}}^{\otimes i}$ and $\mu_{p^{\infty}}^{\otimes i}$ can be
endowed with $G_{S}(F)$-module structures via the diagonal action. Therefore,
we may speak of the Galois cohomology groups
$H^{k}\big{(}G_{S}(F),\mu_{p^{n}}^{\otimes i}\big{)}$ and
$H^{k}\big{(}G_{S}(F),\mu_{p^{\infty}}^{\otimes i}\big{)}$, noting that

$H^{k}\big{(}G_{S}(F),\mu_{p^{\infty}}^{\otimes
i}\big{)}\cong\displaystyle\mathop{\varinjlim}\limits_{n}H^{k}\big{(}G_{S}(F),\mu_{p^{n}}^{\otimes
i}\big{)}.$

In particular, for $1\leq m\leq\infty$, we shall write $\mu_{p^{m}}^{\otimes
i}(L)$ for $H^{0}(G_{S}(L),\mu_{p^{n}}^{\otimes i})=(\mu_{p^{n}}^{\otimes
i})^{G_{S}(L)}$.

Let $S_{p}$ be the set of primes of $F$ above $p$. For a finite extension $L$
of $F$ contained in $F_{S}$, we will write $S_{p}(L)$ for the set of primes of
$L$ above $S_{p}$. If $w$ is a prime in $S_{p}(L)$, we then write $L_{w}$ for
the completion of $L$ at $w$. One has a natural group homomorphism

$\operatorname{Gal}(\overline{L_{w}}/L_{w})\longrightarrow\operatorname{Gal}(\overline{L}/L)\longrightarrow\operatorname{Gal}(F_{S}/L).$

Via this homomorphism, the $i$-fold tensor product $\mu_{p^{m}}^{\otimes i}$
may be viewed as a $\operatorname{Gal}(\overline{L_{w}}/L_{w})$-module for
$1\leq m\leq\infty$, and we then set $\mu_{p^{m}}^{\otimes i}(L_{w})$ to be
$H^{0}(\operatorname{Gal}(\overline{L_{w}}/L_{w}),\mu_{p^{m}}^{\otimes i})$.

The next lemma gives a precise description of these groups.

######  Lemma 3.2.

Suppose that $i\geq 1$. Let $L$ be a finite extension of $F$ contained in
$F_{S}$. Let $\mathcal{L}$ denote either $L$ or $L_{w}$ for some $w\in
S_{p}(L)$. Set $a(\mathcal{L})$ to be the largest integer such that
$\mathcal{L}(\mu_{p})$ contains a primitive $p^{a(\mathcal{L})}$th root of
unity and set $b(i)=v_{p}(i)$. Then we have

$\mu_{p^{\infty}}^{\otimes
i}(\mathcal{L})=\left\\{\begin{array}[]{ll}\mu_{p^{a(\mathcal{L})+b(i)}}^{\otimes
i},&\hbox{if $i\equiv 0$ mod $|\mathcal{L}(\mu_{p}):\mathcal{L}|$,}\\\
1,&\hbox{if $i\not\equiv 0$ mod
$|\mathcal{L}(\mu_{p}):\mathcal{L}|$.}\end{array}\right.$

In particular, one has

$\mu_{p^{n}}^{\otimes
i}(\mathcal{L})=\left\\{\begin{array}[]{ll}\mu_{p^{n}}^{\otimes i},&\hbox{if
$i\equiv 0$ mod $|\mathcal{L}(\mu_{p}):\mathcal{L}|$ and $n\leq
a(\mathcal{L})+b(i)$,}\\\ \mu_{p^{a(\mathcal{L})+b(i)}}^{\otimes i},&\hbox{if
$i\equiv 0$ mod $|\mathcal{L}(\mu_{p}):\mathcal{L}|$ and
$n>a(\mathcal{L})+b(i)$,}\\\ 1,&\hbox{if $i\not\equiv 0$ mod
$|\mathcal{L}(\mu_{p}):\mathcal{L}|$.}\end{array}\right.$

###### Proof.

See [33, Chap. VI, Proposition 2.2]. ∎

######  Remark 3.3.

Note that the final equality is also saying that whenever $i=0$ mod
$|\mathcal{L}(\mu_{p}):\mathcal{L}|$ and $n\leq a(\mathcal{L})+b(i)$,
$\mu_{p^{n}}^{\otimes i}$ is a trivial $\mathcal{G}$-module, where
$\mathcal{G}$ is $G_{S}(L)$ or $\operatorname{Gal}(\overline{L_{w}}/L_{w})$
accordingly to $\mathcal{L}$ being $L$ or $L_{w}$.

Let $L$ be a finite extension of $F$. To ease the notation slightly, we shall
write $\operatorname{Cl}_{S}(\mathcal{O}_{L})$ for the $S(L)$-class group
$\operatorname{Cl}_{S(L)}(\mathcal{O}_{L})$. We then write $A_{L}$ (resp.
$A_{L}^{S}$) for the Sylow $p$-subgroup of
$\operatorname{Cl}(\mathcal{O}_{L})$ (resp., Sylow $p$-subgroup of
$\operatorname{Cl}_{S}(\mathcal{O}_{L})$). The following exact sequence
relates this class group with the cohomology group
$H^{2}(G_{S}(L),\mu_{p^{n}})$.

######  Proposition 3.4.

Let $L$ be a finite extension of $F$ contained in $F_{S}$. Then for every
$n\geq 1$, one has the following exact sequence

$0\longrightarrow A_{L}^{S}/p^{n}\longrightarrow
H^{2}(G_{S}(L),\mu_{p^{n}})\longrightarrow\bigoplus_{w\in
S_{p}(L)}\mathbb{Z}/p^{n}\mathbb{Z}\longrightarrow\mathbb{Z}/p^{n}\mathbb{Z}\longrightarrow
0.$

###### Proof.

See [23, Satz 4] or [21, Proposition 8.3.11]. ∎

###  3.2 A brief interlude on étale cohomology

On the other hand, we can view $\mu_{p^{n}}^{\otimes i}$ as an étale sheaf
over the scheme $\mathrm{Spec}(\mathcal{O}_{F,S})$ in the sense of [19, Chap.
II], and consider the étale cohomology groups
$H^{k}_{\acute{e}t}\big{(}\mathrm{Spec}(\mathcal{O}_{F,S}),\mu_{p^{n}}^{\otimes
i}\big{)}$. By [19, Chap. II, Proposition 2.9], the latter is isomorphic to
the Galois cohomology groups $H^{k}\big{(}G_{S}(F),\mu_{p^{n}}^{\otimes
i}\big{)}$. Writing
$\mathbb{Z}_{p}(i)=\displaystyle\mathop{\varprojlim}\limits_{n}\mu_{p^{n}}^{\otimes
i}$ and taking inverse limit, we therefore obtain

$\displaystyle\mathop{\varprojlim}\limits_{n}H^{k}_{\acute{e}t}\big{(}\mathrm{Spec}(\mathcal{O}_{F,S}),\mu_{p^{n}}^{\otimes
i}\big{)}\cong\displaystyle\mathop{\varprojlim}\limits_{n}H^{k}\big{(}G_{S}(F),\mu_{p^{\infty}}^{\otimes
i}\big{)}\cong H^{k}_{\mathrm{cts}}\big{(}G_{S}(F),\mathbb{Z}_{p}(i)\big{)},$

where $H^{k}_{\mathrm{cts}}(~{},~{})$ is the continuous cohomology group of
Tate (see [21, Chap 2, §7]), and where the second isomorphism is a consequence
of [21, Corollary 2.7.6 and Theorem 8.3.20(i)]. To simplify notation, we shall
write
$H^{k}_{\acute{e}t}\big{(}\mathcal{O}_{F,S},\mathbb{Z}_{p}(i)\big{)}=\displaystyle\mathop{\varprojlim}\limits_{n}H^{k}_{\acute{e}t}\big{(}\mathrm{Spec}(\mathcal{O}_{F,S}),\mu_{p^{n}}^{\otimes
i}\big{)}$.

###  3.3 Algebraic $K$-theory

We come to the algebraic $K$-theoretical aspects. As before, $p$ will always
denote an odd prime. For a ring $R$ with identity, $K_{n}(R)$ will always
denote the algebraic $K$-groups of $R$ in the sense of Quillen [27, 28] (also
see [16, 33]). In particular, we are interested in understanding
$K_{2i}(\mathcal{O}_{F})$ for $i\geq 1$, where it is well-known that these
groups are finite by the results of Quillen (see [28, 3, 8]).

In [24], Soulé connected the higher $K$-groups with étale cohomology groups
via the $p$-adic Chern class maps

$\mathrm{ch}_{i,k}^{(p)}:K_{2i+2-k}(\mathcal{O}_{F})\otimes\mathbb{Z}_{p}\longrightarrow
H^{k}_{\acute{e}t}\left(\mathcal{O}_{F,S},\mathbb{Z}_{p}(i+1)\right)$

for $i\geq 1$ and $k=1,2$. (For the precise definition of these Chern class
maps, we refer readers to loc. cit.) The famed Quillen-Lichtenbaum Conjecture
asserted that these maps are isomorphisms (for instance, see [16, 33] for the
history). Thanks to the gallant efforts of many, this prediction is now known
to be true.

######  Theorem 3.5.

For an odd prime $p$, the $p$-adic Chern class maps are isomorphisms for
$i\geq 1$ and $k=1,2$.

###### Proof.

Soulé first proved that these maps are surjective (see [24, Théorème 6(iii)];
also see the work of Dwyer and Friedlander [7, Theorem 8.7]). It is folklore
(for instance, see [16, Theorem 2.7]) that the asserted bijectivity is a
consequence of the norm residue isomorphism theorem (previously also known as
the Bloch-Kato(-Milnor) conjecture; see [2, 20] for details). This said norm
residue isomorphism theorem was proved by the deep work of Rost and Voevodsky
[31] (also see [22]) with the aid of a patch from Weibel [32]. ∎

######  Corollary 3.6.

Let $p$ be an odd prime. For $i\geq 1$, we have

$K_{2i}(\mathcal{O}_{F})[p^{\infty}]\cong
H^{2}_{\mathrm{cts}}\big{(}G_{S}(F),\mathbb{Z}_{p}(i+1)\big{)}$

Furthermore, for each $n\geq 1$, we have

$K_{2i}(\mathcal{O}_{F})/p^{n}\cong H^{2}\big{(}G_{S}(F),\mu_{p^{n}}^{\otimes
i+1}\big{)}.$

###### Proof.

Since the group $K_{2i}(\mathcal{O}_{F})$ is finite, it follows that
$K_{2i}(\mathcal{O}_{F})[p^{\infty}]\cong
K_{2i}(\mathcal{O}_{F})\otimes\mathbb{Z}_{p}$. By the preceding theorem, the
latter is isomorphic to
$H^{2}_{\acute{e}t}\big{(}\mathcal{O}_{F,S},\mathbb{Z}_{p}(i+1)\big{)}$ which
identifies with the corresponding continuous Tate cohomology group as noted in
Subsection 3.2. The final isomorphism now follows from considering the long
exact sequence of the $G_{S}(F)$-cohomology of the short exact sequence

$0\longrightarrow\mathbb{Z}_{p}(i+1)\stackrel{{\scriptstyle
p^{n}}}{{\longrightarrow}}\mathbb{Z}_{p}(i+1)\longrightarrow\mu_{p^{n}}^{\otimes
i+1}\longrightarrow 0$

and noting that $H^{3}_{\mathrm{cts}}\big{(}G_{S}(F),-\big{)}=0$ (cf. [21,
Proposition 10.11.3]). ∎

######  Remark 3.7.

We should mention that the conclusions of Theorem 3.5 and Corollary 3.6 are
not true in general when $p=2$. As the current paper is concerned with the
case of an odd prime $p$, we shall say nothing more on this but refer the
interested readers to [22] for details.

##  4 Main results

###  4.1 Twist

As before, $p$ is an odd prime. Let $m$ be a positive integer. Once and for
all, we fix a primitive $p^{m}$th root of unity $\zeta$ in $\bar{\mathbb{Q}}$.
Under such a choice, denote by $\sigma_{c}$ the element of
$\operatorname{Gal}(\mathbb{Q}(\mu_{p^{m}})/\mathbb{Q})$ which is defined by
$\sigma_{c}(\zeta)=\zeta^{c}$. This assignment in turn induces an isomorphism
$\operatorname{Gal}(\mathbb{Q}(\mu_{p^{m}})/\mathbb{Q})\cong(\mathbb{Z}/p^{m}\mathbb{Z})^{\times}$
of groups.

For a number field $F$, we let $\Delta$ denote the maximal subgroup of
$\operatorname{Gal}(F(\mu_{p^{m}})/F)$ with order coprime to $p$. In other
words, $\Delta\cong\operatorname{Gal}(F(\mu_{p})/F)$. (Of course, the group
$\Delta$ may be trivial, for instance, if $F$ contains $\mu_{p}$.) We shall
write $d=|\Delta|$. The Galois group $\operatorname{Gal}(F(\mu_{p^{m}})/F)$
may be identified with a subgroup of
$\operatorname{Gal}(\mathbb{Q}(\mu_{p^{m}})/\mathbb{Q})$, and under this
identification, the group $\Delta$ can be identified as a subgroup of
$(\mathbb{Z}/p\mathbb{Z})^{\times}$. For our subsequent discussion, we require
an explicit description of this embedding of $\Delta$ into
$(\mathbb{Z}/p\mathbb{Z})^{\times}$. Fix a generator $g$ of
$(\mathbb{Z}/p^{m}\mathbb{Z})^{\times}$, and write
$\sigma:=(\sigma_{g})^{p^{m-1}}$. Under these settings, the elements of the
group $\Delta$ may be identified with elements of the form
$\sigma^{k(p-1)/d}$, where $k=0,1,...,d-1$. Under the surjection
$(\mathbb{Z}/p^{m}\mathbb{Z})^{\times}\twoheadrightarrow(\mathbb{Z}/p\mathbb{Z})^{\times}$,
the element $g$ is mapped to a generator of
$(\mathbb{Z}/p\mathbb{Z})^{\times}$ which by abuse of notation is still
denoted as $g$. Therefore, correspondingly, we have $g^{(p-1)/d}$ being a
generator of the subgroup of $(\mathbb{Z}/p\mathbb{Z})^{\times}$ which is the
image of $\Delta$ under the embedding of $\Delta$ into
$(\mathbb{Z}/p\mathbb{Z})^{\times}$.

Let $\omega$ be the Teichmüller character of the group
$(\mathbb{Z}/p\mathbb{Z})^{\times}$. For a given integer $j$, we define

$\varepsilon_{j}:=\varepsilon_{j}(F):=\frac{1}{d}\sum_{k=0}^{d-1}\omega(g)^{jk(p-1)/d}\sigma^{-k(p-1)/d}$

which lives in
$\mathbb{Z}_{p}[\Delta]\subseteq\mathbb{Z}_{p}[\operatorname{Gal}(F(\mu_{p^{m}})/F)]$.
One can check easily that $\varepsilon_{j}=\varepsilon_{j^{\prime}}$ whenever
$j\equiv j^{\prime}$ (mod $|\Delta|$), and that
$\varepsilon_{0},...,\varepsilon_{d-1}$ forms a collection of primitive
idempotents of the group ring $\mathbb{Z}_{p}[\Delta]$.

######  Lemma 4.1.

Let $M$ be a $\mathbb{Z}_{p}[\Delta]$-module. Then the following statements
are valid.

  1. $(i)$

$M^{\Delta}=\varepsilon_{0}M$.

  2. $(ii)$

For every $i\geq 1$, we have $\big{(}\mu_{p^{m}}^{\otimes i}\otimes
M\big{)}^{\Delta}=\mu_{p^{m}}^{\otimes i}\otimes\varepsilon_{-i}M$.

###### Proof.

The first assertion is immediate from the definition of $\varepsilon_{0}$. For
the second, it suffices to show that

$\varepsilon_{j+1}(\mu_{p^{m}}\otimes M)=\mu_{p^{m}}\otimes\varepsilon_{j}M$

for every $j$. Indeed, the conclusion of the lemma follows from a recursive
application of the said equality. Via the natural surjection
$(\mathbb{Z}/p^{m}\mathbb{Z})^{\times}\twoheadrightarrow(\mathbb{Z}/p\mathbb{Z})^{\times}$,
we may also view $\omega$ as a character of
$(\mathbb{Z}/p^{m}\mathbb{Z})^{\times}$. It then follows that
$\omega(g)^{p-1}=1$ and $\omega(g)=g$ mod $p$, where the latter in turn
implies that $\omega(g)=\omega(g)^{p^{m-1}}=g^{p^{m-1}}$ mod $p^{m-1}$.
Consequently, we have

$\sigma(\zeta)=\zeta^{g^{p^{m-1}}}=\zeta^{\omega(g)}.$

Let $x\in M$. Then we have

$\varepsilon_{j+1}(\zeta\otimes
x)=\frac{1}{d}\sum_{k=0}^{d-1}\omega(g)^{(j+1)k(p-1)/d}\sigma^{-k(p-1)/d}(\zeta\otimes
x)$
$=\frac{1}{d}\sum_{k=0}^{d-1}\omega(g)^{jk(p-1)/d}\sigma^{-k(p-1)/d}(\zeta^{\omega(g)^{k(p-1)/d}}\otimes
x)$
$=\frac{1}{d}\sum_{k=0}^{d-1}\omega(g)^{jk(p-1)/d}(\zeta\otimes\sigma^{-k(p-1)/d}x)=\zeta\otimes\varepsilon_{j}x.$

This establishes our claim and the proof of the lemma is thus complete. ∎

In what follows, we will study the $p^{n}$-rank of $K_{2i}(\mathcal{O}_{F})$
for a given number field $F$ and integer $i\geq 1$. Let $a:=a(F)$ be the
largest integer such that $F(\mu_{p})=F(\mu_{p^{a}})$. Set $b=v_{p}(i)$. In
other words, $b$ is the largest integer such that $p^{b}$ divides $i$. We
first consider the case when $n\leq a+b$.

###  4.2 $p^{n}$-rank for $n\leq a+b$

For a given integer $i\geq 1$, let $S_{p}^{(i)}$ be the set of primes $v$ in
$S_{p}$ such that $i$ is divisible by $|\Delta_{v}|$, where $\Delta_{v}$ is
the decomposition group of $\Delta$ at $v$. In particular, $S_{p}^{(1)}$ is
the set of primes $v$ in $S_{p}$ which splits completely in $F(\mu_{p})/F$. We
can now state and prove the following.

######  Theorem 4.2.

Suppose that $n\leq a+b$.

  * $(i)$

If $i\equiv 0$ mod $|F(\mu_{p}):F|$, we have the following exact sequence

$0\longrightarrow\mu_{p^{n}}^{\otimes i}\otimes A^{S}_{F}\longrightarrow
K_{2i}(\mathcal{O}_{F})/p^{n}\longrightarrow\bigoplus_{v\in
S_{p}}\mu_{p^{n}}^{\otimes i}\longrightarrow\mu_{p^{n}}^{\otimes
i}\longrightarrow 0.$

  * $(ii)$

If $i\not\equiv 0$ mod $|F(\mu_{p}):F|$, we have the following exact sequence

$0\longrightarrow\mu_{p^{n}}^{\otimes
i}\otimes\varepsilon_{-i}A^{S}_{F(\mu_{p})}\longrightarrow
K_{2i}(\mathcal{O}_{F})/p^{n}\longrightarrow\bigoplus_{v\in
S_{p}^{(i)}}\mu_{p^{n}}^{\otimes i}\longrightarrow 0.$

###### Proof.

We first consider the situation when $i\equiv 0$ mod $|F(\mu_{p}):F|$. From
Proposition 3.4, we have the following exact sequence

$0\longrightarrow A^{S}_{F}/p^{n}\longrightarrow
H^{2}(G_{S}(F),\mu_{p^{n}})\longrightarrow\bigoplus_{v\in
S_{p}}\mathbb{Z}/p^{n}\mathbb{Z}\longrightarrow\mathbb{Z}/p^{n}\mathbb{Z}\longrightarrow
0.$

Applying $\mu_{p^{n}}^{\otimes i}\otimes-$, we obtain

$0\longrightarrow\mu_{p^{n}}^{\otimes i}\otimes
A^{S}_{F}\longrightarrow\mu_{p^{n}}^{\otimes i}\otimes
H^{2}(G_{S}(F),\mu_{p^{n}})\longrightarrow\bigoplus_{v\in
S_{p}}\mu_{p^{n}}^{\otimes i}\longrightarrow\mu_{p^{n}}^{\otimes
i}\longrightarrow 0.$

Since $i\equiv 0$ mod $|F(\mu_{p}):F|$, the group $G_{S}(F)$ acts trivially on
$\mu_{p^{n}}^{\otimes i}$, and so we have

$\mu_{p^{n}}^{\otimes i}\otimes H^{2}(G_{S}(F),\mu_{p^{n}})\cong
H^{2}(G_{S}(F),\mu_{p^{n}}^{\otimes i+1}).$

By Corollary 3.6, the latter is $K_{2i}(\mathcal{O}_{F})/p^{n}$. This proves
(i).

Now suppose that $i\not\equiv 0$ mod $|F(\mu_{p}):F|$. In this case, we have
$G_{S}(F(\mu_{p}))$ acting trivially on $\mu_{p^{n}}^{\otimes i}$ by Lemma
3.1, since $a(F(\mu_{p}))=a$. By a similar argument as above, we obtain the
following exact sequence

$0\longrightarrow\mu_{p^{n}}^{\otimes i}\otimes
A^{S}_{F(\mu_{p})}\longrightarrow H^{2}(G_{S}(F(\mu_{p})),\mu_{p^{n}}^{\otimes
i+1})\longrightarrow\bigoplus_{w\in S_{p}(F(\mu_{p}))}\mu_{p^{n}}^{\otimes
i}\longrightarrow\mu_{p^{n}}^{\otimes i}\longrightarrow 0.$

Taking $\Delta$-invariant, we have

$0\longrightarrow\big{(}\mu_{p^{n}}^{\otimes i}\otimes
A^{S}_{F(\mu_{p})})\big{)}^{\Delta}\longrightarrow
H^{2}(G_{S}(F(\mu_{p})),\mu_{p^{n}}^{\otimes i+1})^{\Delta}\\\
\longrightarrow\Big{(}\bigoplus_{w\in S_{p}(F(\mu_{p}))}\mu_{p^{n}}^{\otimes
i}\Big{)}^{\Delta}\longrightarrow\big{(}\mu_{p^{n}}^{\otimes
i}\big{)}^{\Delta}\longrightarrow 0.$

Note that the above sequence is still exact, as $|\Delta|$ is coprime to $p$.
In view of this latter coprime property, one may also apply Lemma 3.1 to
conclude that

$H^{2}\big{(}G_{S}(F(\mu_{p})),\mu_{p^{n}}^{\otimes i+1}\big{)}^{\Delta}\cong
H^{2}(G_{S}(F),\mu_{p^{n}}^{\otimes i+1}),$

where the latter is $K_{2i}(\mathcal{O}_{F})/p^{n}$ by Corollary 3.6. The
first term in the exact sequence is

$\mu_{p^{n}}^{\otimes i}\otimes\varepsilon_{-i}A^{S}_{F(\mu_{p})}$

by Lemma 4.1. On the other hand, as $i\not\equiv 0$ mod $|F(\mu_{p}):F|$, one
has $\big{(}\mu_{p^{n}}^{\otimes i}\big{)}^{\Delta}=0$ by Lemma 3.1. For the
remaining local terms, we note that

$\Big{(}\bigoplus_{w\in S_{p}(F(\mu_{p}))}\mu_{p^{n}}^{\otimes
i}\Big{)}^{\Delta}=\bigoplus_{v\in S_{p}}\Big{(}\bigoplus_{w\mid
v}\mu_{p^{n}}^{\otimes i}\Big{)}^{\Delta}\cong\bigoplus_{v\in
S_{p}}\big{(}\mu_{p^{n}}^{\otimes i}\big{)}^{\Delta_{v}},$

where the second isomorphism follows from Shapiro’s lemma. By another
application of Lemma 3.1, one then has

$(\mu_{p^{n}}^{\otimes
i}\big{)}^{\Delta_{v}}=\left\\{\begin{array}[]{ll}\mu_{p^{n}}^{\otimes
i},&\hbox{if $i$ is divisible by $|\Delta_{v}|$,}\\\
0,&\hbox{otherwise.}\end{array}\right.$

Combining all the above observations, we obtain the conclusion of (ii). ∎

We discuss some consequences of the preceding theorem.

######  Corollary 4.3.

Suppose that $n\leq a+b$. Then

$r_{p^{n}}\big{(}K_{2i}(\mathcal{O}_{F})\big{)}=\left\\{\begin{array}[]{ll}r_{p^{n}}\big{(}A^{S}_{F}\big{)}+|S_{p}|-1,&\hbox{if
$i\equiv 0$ mod $|F(\mu_{p}):F|$,}\\\
r_{p^{n}}\big{(}\varepsilon_{-i}A^{S}_{F(\mu_{p})}\big{)}+|S_{p}^{(i)}|,&\hbox{if
$i\not\equiv 0$ mod $|F(\mu_{p}):F|$.}\end{array}\right.$

###### Proof.

Suppose that $i=0$ mod $|F(\mu_{p}):F|$. Set $N$ to be the kernel of the map
$\bigoplus_{v\in S_{p}}\mu_{p^{n}}^{\otimes
i}\longrightarrow\mu_{p^{n}}^{\otimes i}$. Then it follows from Theorem 4.2(i)
that we have a short exact sequence

$0\longrightarrow\mu_{p^{n}}^{\otimes i}\otimes A^{S}_{F}\longrightarrow
K_{2i}(\mathcal{O}_{F})/p^{n}\longrightarrow N\longrightarrow 0$

of $\mathbb{Z}/p^{n}\mathbb{Z}$-modules. Plainly, $N$ is a free
$\mathbb{Z}/p^{n}\mathbb{Z}$-module of rank $|S_{p}|-1$, and so the above
short exact sequence splits, which in turn supplies an isomorphism

$K_{2i}(\mathcal{O}_{F})/p^{n}\cong\big{(}\mu_{p^{n}}^{\otimes i}\otimes
A^{S}_{F}\big{)}\oplus N$

of $\mathbb{Z}/p^{n}\mathbb{Z}$-modules. The asserted $p^{n}$-rank of
$K_{2i}(\mathcal{O}_{F})$ is now an immediate consequence of this.

Suppose that $i\neq 0$ $($mod $|\Delta|)$. By Theorem 4.2, we have a short
exact sequence

$0\longrightarrow\mu_{p^{n}}^{\otimes
i}\otimes\varepsilon_{-i}A^{S}_{F(\mu_{p})}\longrightarrow
K_{2i}(\mathcal{O}_{F})/p^{n}\longrightarrow\bigoplus_{v\in
S_{p}^{(i)}}\mu_{p^{n}}^{\otimes i}\longrightarrow 0,$

where $\bigoplus_{v\in S_{p}^{(i)}}\mu_{p^{n}}^{\otimes i}$ is a free
$\mathbb{Z}/p^{n}\mathbb{Z}$-module of rank $|S_{p}^{(i)}|$. Therefore, the
sequence splits and the asserted $p^{n}$-rank of $K_{2i}(\mathcal{O}_{F})$
follows from this. ∎

We record the following special case of the preceding corollary which
generalizes a previous result of Kolster (see [14, Corollary 1.8] and [15,
Corollary 3.4]), where a formula for the $p^{n}$-rank of $K_{2}$ is obtained.

######  Corollary 4.4.

Suppose $n\leq a+b$, $|F(\mu_{p}):F|=2$ and $i$ is an odd integer. Then we
have

$r_{p^{n}}(K_{2i}(\mathcal{O}_{F}))=r_{p^{n}}\big{(}\varepsilon_{1}A^{S}_{F(\mu_{p})}\big{)}+|S_{p}^{(1)}|.$

###  4.3 $p^{n}$-rank for $n>a+b$

We now consider the situation when $n$ is greater than $a+b$.

######  Theorem 4.5.

If $n>a+b$, we have the following exact sequence

$0\longrightarrow\big{(}\mu_{p^{n}}^{\otimes i}\otimes
A^{S}_{E}\big{)}_{G}\longrightarrow
K_{2i}(\mathcal{O}_{F})/p^{n}\longrightarrow\bigoplus_{v\in
S_{p}}\mu_{p^{n}}^{\otimes i}(F_{v})\longrightarrow\mu_{p^{n}}^{\otimes
i}(F)\longrightarrow 0$

where $E:=E_{n}:=F(\mu_{p^{n-b}})$ and $G=\operatorname{Gal}(E/F)$.

In preparation of the proof of the theorem, we shall first establish the
following lemma which generalizes an observation made by Keune in [12, Lemma
6.5]. We note that our approach of proof differs from that of Keune.

######  Lemma 4.6.

Retain the settings of Theorem 4.5. Then $\mu_{p^{n}}^{\otimes i}$ is a
cohomologically trivial $H$-module for every subgroup $H$ of $G$.

###### Proof.

By replacing $F$ by $E^{H}$, it suffices to prove that $\mu_{p^{n}}^{\otimes
i}$ is a cohomologically trivial $G$-module. Set
$\Gamma=\operatorname{Gal}(F(\mu_{p^{\infty}})/F)$ and
$\Gamma_{n}=\operatorname{Gal}(F(\mu_{p^{\infty}})/E)$. Consider the following
spectral sequence

$H^{r}\big{(}G,H^{s}(\Gamma_{n},\mu_{p^{\infty}}^{\otimes
i})\big{)}\Longrightarrow H^{r+s}(\Gamma,\mu_{p^{\infty}}^{\otimes i}).$

Since the groups $\Gamma_{n}$ and $\Gamma$ have $p$-cohomological dimension
one, one has

$H^{t}(\Gamma_{n},\mu_{p^{\infty}}^{\otimes
i})=H^{t}(\Gamma,\mu_{p^{\infty}}^{\otimes i})=0$

for $t\geq 2$. On the other hand, Tate Lemma [29] tells us that

$H^{1}(\Gamma_{n},\mu_{p^{\infty}}^{\otimes
i})=H^{1}(\Gamma,\mu_{p^{\infty}}^{\otimes i})=0.$

Hence the spectral sequence degenerates yielding

$H^{r}\big{(}G,H^{0}(\Gamma_{n},\mu_{p^{\infty}}^{\otimes i})\big{)}=0$

for every $r\geq 1$. But one also has
$H^{0}(\Gamma_{n},\mu_{p^{\infty}}^{\otimes i})=\mu_{p^{n}}^{\otimes i}$ by
Lemma 3.1. This completes the proof of the lemma. ∎

We can now give the proof of Theorem 4.5.

###### Proof of Theorem 4.5.

In view that $G_{S}(E)$ acts trivially on $\mu_{p^{n}}^{\otimes i}$, we may
apply a similar argument as in the beginning of the proof of Theorem 4.2 to
obtain the following exact sequence

$0\longrightarrow\mu_{p^{n}}^{\otimes i}\otimes A^{S}_{E}\longrightarrow
H^{2}(G_{S}(E),\mu_{p^{n}}^{\otimes i+1})\longrightarrow\bigoplus_{w\in
S_{p}(E)}\mu_{p^{n}}^{\otimes i}\longrightarrow\mu_{p^{n}}^{\otimes
i}\longrightarrow 0.$

By Lemma 4.6, we see that $\mu_{p^{n}}^{\otimes i}$ is a cohomologically
trivial $G$-module. We now proceed to show that $\bigoplus_{w\in
S_{p}(E)}\mu_{p^{n}}^{\otimes i}$ is also a cohomologically trivial
$G$-module. Observe that

$H^{r}\Big{(}G,\bigoplus_{w\in S_{p}(E)}\mu_{p^{n}}^{\otimes
i}\Big{)}\cong\bigoplus_{v\in
S_{p}}H^{r}\Big{(}G,\bigoplus_{w|v}\mu_{p^{n}}^{\otimes
i}\Big{)}\cong\bigoplus_{v\in S_{p}}H^{r}\Big{(}G_{v},\mu_{p^{n}}^{\otimes
i}\Big{)},$

where the second isomorphism is a consequence of Shapiro’s lemma. Since
$G_{v}$ may be identified as a subgroup of $G$, we may apply Lemma 4.6 to
conclude that $H^{r}\Big{(}G_{v},\mu_{p^{n}}^{\otimes i}\Big{)}=0$ for $r\geq
1$. Hence this shows that $\bigoplus_{w\in S_{p}(E)}\mu_{p^{n}}^{\otimes i}$
is a cohomologically trivial $G$-module. Set
$Y:=\ker\Big{(}\displaystyle\bigoplus_{w\in S_{p}(E)}\mu_{p^{n}}^{\otimes
i}\longrightarrow\mu_{p^{n}}^{\otimes i}\Big{)}$. A straightforward
cohomological consideration shows that $Y$ is also a cohomologically trivial
$G$-module. Therefore, after taking $-_{G}$ functor, we have the following
short exact sequences

$0\longrightarrow\Big{(}\mu_{p^{n}}^{\otimes i}\otimes
A^{S}_{E}\Big{)}_{G}\longrightarrow\Big{(}H^{2}(G_{S}(E),\mu_{p^{n}}^{\otimes
i+1})\Big{)}_{G}\longrightarrow Y_{G}\longrightarrow 0,$ $0\longrightarrow
Y_{G}\longrightarrow\Big{(}\bigoplus_{w\in S_{p}(E)}\mu_{p^{n}}^{\otimes
i}\Big{)}_{G}\longrightarrow\Big{(}\mu_{p^{n}}^{\otimes
i}\Big{)}_{G}\longrightarrow 0.$

Splicing these exact sequences, we obtain

$0\longrightarrow\Big{(}\mu_{p^{n}}^{\otimes i}\otimes
A^{S}_{E}\Big{)}_{G}\longrightarrow\Big{(}H^{2}(G_{S}(E),\mu_{p^{n}}^{\otimes
i+1})\Big{)}_{G}\longrightarrow\Big{(}\bigoplus_{w\in
S_{p}(E)}\mu_{p^{n}}^{\otimes
i}\Big{)}_{G}\longrightarrow\Big{(}\mu_{p^{n}}^{\otimes
i}\Big{)}_{G}\longrightarrow 0.$

By Lemma 3.1, $\Big{(}H^{2}(G_{S}(E),\mu_{p^{n}}^{\otimes
i+1})\Big{)}_{G}\cong H^{2}(G_{S}(F),\mu_{p^{n}}^{\otimes i+1})$ which is
precisely $K_{2i}(\mathcal{O}_{F})/p^{n}$ by Corollary 3.6. Finally, for every
cohomologically trivial $G$-module $N$, one has a natural isomorphism
$N_{G}\cong N^{G}$ induced by the norm map. Therefore, for the rightmost two
terms, we may switch $-_{G}$ to $-^{G}$. Putting all these observations
together, we obtain the conclusion of our theorem. ∎

##  5 Quadratic fields

We specialize to the quadratic fields. Let $m$ be a squarefree integer
$\neq\pm 1,(-1)^{(p-1)/2}p$. Set $F=\mathbb{Q}(\sqrt{m})$. Note that in this
sitation, one has $a=a(F)=1$. As before, we continue to write $b=v_{p}(i)$.

We begin with the case of small $n$.

######  Proposition 5.1.

Suppose that $n\leq 1+b$. Then the following statements are valid.

  * $(i)$

If $i\equiv 0$ mod $p-1$, then

$r_{p^{n}}\big{(}K_{2i}(\mathcal{O}_{F})\big{)}=\left\\{\begin{array}[]{ll}r_{p^{n}}\big{(}A^{S}_{F}\big{)}+1,&\hbox{if
$p$ splits in $F/\mathbb{Q}$,}\\\
r_{p^{n}}\big{(}A^{S}_{F}\big{)},&\hbox{otherwise.}\end{array}\right..$

  * $(ii)$

If $i\not\equiv 0$ mod $p-1$, then we have

$r_{p^{n}}(K_{2i}(\mathcal{O}_{F}))\\\
=\left\\{\begin{array}[]{ll}r_{p^{n}}\big{(}\varepsilon_{-i}A^{S}_{F(\mu_{p})}\big{)}+1,&\hbox{if
$i\equiv 0$ mod $(p-1)/2$ and $m=(-1)^{(p-1)/2}pm_{1}$}\\\ &\hbox{ with $p$
being split in $\mathbb{Q}(\sqrt{m_{1}})/\mathbb{Q}$,}\\\
r_{p^{n}}\big{(}\varepsilon_{-i}A^{S}_{F(\mu_{p})}\big{)},&\hbox{otherwise.}\end{array}\right.$

###### Proof.

(i) Since $|S_{p}|=2$ or $1$ accordingly to the prime $p$ splitting in
$F/\mathbb{Q}$ or not, the asserted $p^{n}$-rank formula is a consequence of
Corollary 4.3.

(ii) From Corollary 4.3, one has

$r_{p^{n}}(K_{2i}(\mathcal{O}_{F}))=r_{p^{n}}\big{(}\varepsilon_{-i}A^{S}_{F(\mu_{p})}\big{)}+|S^{(i)}|.$

From the discussion in [36, Section 3], we see that for a prime $v\in S_{p}$,
$|F_{v}(\mu_{p}):F_{v}|=(p-1)/2$ precisely when the prime $p$ is both ramified
in $F/\mathbb{Q}$ and split in $\mathbb{Q}(\sqrt{m_{1}})/\mathbb{Q}$, where
$m=(-1)^{(p-1)/2}pm_{1}$. The discussion in [36, Section 3] also tells us that
in all other cases, one has $|F_{v}(\mu_{p}):F_{v}|=p-1$. Hence we have
$|S_{p}^{(i)}|=1$ when $i$ is divisible by $(p-1)/2$, and
$m=(-1)^{(p-1)/2}pm_{1}$ with $p$ splitting in
$\mathbb{Q}(\sqrt{m_{1}})/\mathbb{Q}$, and $|S_{p}^{(i)}|=0$ otherwise. This
yields the asserted conclusion. ∎

For large $n$, we may apply Theorem 4.5 to obtain the following. Since the
argument is quite similar to that of the preceding proof, we shall omit the
proof.

######  Proposition 5.2.

Suppose that $n>1+b$. Set $E=F(\mu_{p^{n-b}})$ and
$G=\operatorname{Gal}(E/F)$. Then the following statements are valid.

  * $(i)$

Suppose that $i\equiv 0$ mod $p-1$.

    * $(a)$

If $p$ is non-split in $F/\mathbb{Q}$, then we have the following exact
sequence

$\big{(}\mu_{p^{n}}^{\otimes i}\otimes A^{S}_{E}\big{)}_{G}\cong
K_{2i}(\mathcal{O}_{F})/p^{n}$

and therefore

$r_{p^{n}}\big{(}K_{2i}(\mathcal{O}_{F})\big{)}=r_{p^{n}}\Big{(}\big{(}\mu_{p^{n}}^{\otimes
i}\otimes A^{S}_{E}\big{)}_{G}\Big{)}.$

    * $(b)$

If $p$ splits in $F/\mathbb{Q}$, then we have

$0\longrightarrow\big{(}\mu_{p^{n}}^{\otimes i}\otimes
A^{S}_{E}\big{)}_{G}\longrightarrow
K_{2i}(\mathcal{O}_{F})/p^{n}\longrightarrow\mu_{p^{1+b}}^{\otimes
i}\longrightarrow 0.$

  * $(ii)$

Suppose that $i\not\equiv 0$ mod $p-1$, then we have

$K_{2i}(\mathcal{O}_{F})/p^{n}\cong\big{(}\mu_{p^{n}}^{\otimes i}\otimes
A^{S}_{E}\big{)}_{G}$

except when one has $i\equiv 0$ mod $(p-1)/2$ and $m=(-1)^{(p-1)/2}pm_{1}$
with $p$ being split in $\mathbb{Q}(\sqrt{m_{1}})/\mathbb{Q}$. In this
exceptional case, one instead has a short exact sequence

$0\longrightarrow\big{(}\mu_{p^{n}}^{\otimes i}\otimes
A^{S}_{E}\big{)}_{G}\longrightarrow
K_{2i}(\mathcal{O}_{F})/p^{n}\longrightarrow\mu_{p^{1+b}}^{\otimes
i}\longrightarrow 0.$

##  6 On Kummer’s criterion

In this final section, $F$ will always denote a totally real field. We let
$\zeta_{p}$ denote a fixed primitive $p$-th root of unity, and set
$d=|F(\zeta_{p}):F|$. The following theorem is the main result of this
section.

######  Theorem 6.1.

Suppose that $F$ is a totally real field such that all the primes in
$S_{p}\big{(}F(\zeta_{p}+\zeta_{p}^{-1})\big{)}$ do not split in
$F(\zeta_{p})$. Then the following statements are equivalent.

  * $(1)$

The class number of $F(\zeta_{p})$ is divisible by $p$.

  * $(2)$

The prime $p$ divides the order of $K_{2i}(\mathcal{O}_{F})$ for some $1\leq
i\leq d-1$ with $i$ being odd.

######  Remark 6.2.

Before giving a proof of the above theorem, we explain how the said theorem is
equivalent to Kummer’s criterion in the sense of [10, Theorem 1] and [13,
Theorem 1], and thus, can be thought as an algebraic $K$-theoretical
formulation of it. Write $|~{}|_{p}$ for the $p$-adic norm which is normalized
with $|p|_{p}=1/p$. Denote by $\zeta_{F}(s)$ the Dedekind zeta function of
$F$. For $a,b\in\mathbb{Q}-\\{0\\}$, we write $a\sim_{p}b$ if $a/b$ is a
$p$-unit. For each odd integer $i$ with $1\leq i\leq d-1$, it follows from the
main conjecture of Iwasawa as proven by Wiles [34] that

$\zeta_{F}\big{(}1-(i+1)\big{)}\sim_{p}\frac{|K_{2i}(\mathcal{O}_{F})|}{|K_{2i+1}(\mathcal{O}_{F})|}.$

(Strictly speaking, the theorem of Wiles will only yield

$\zeta_{F}\big{(}1-(i+1)\big{)}\sim_{p}\frac{|H^{2}(G_{S}(F),\mathbb{Z}_{p}(i+1)|}{|H^{1}(G_{S}(F),\mathbb{Z}_{p}(i+1)|};$

see the paper of Báyer and Neukirch [1]. But we now know that this
cohomological version is equivalent to the $K$-theoretical version as stated
above in view of the work of Rost-Voevodsky [31]).

From [33, Chap. VI, Theorem 9.5], we see that
$\big{|}K_{2i+1}(\mathcal{O}_{F})\big{|}\sim_{p}w^{(p)}_{i+1}(F)$, where
$w_{j}^{(p)}(F)$ is the order of $\mu_{p^{\infty}}^{\otimes
j}(F)=(\mu_{p^{\infty}}^{\otimes j})^{\operatorname{Gal}(\bar{F}/F)}$. Lemma
3.2 then tells us that

$w^{(p)}_{i+1}(F)=\left\\{\begin{array}[]{ll}1,&\hbox{$1\leq i<d-1$, $i$
odd}\\\ p^{a(F)},&\hbox{$i=d-1$.}\end{array}\right.$

Therefore, upon combining these observations, we obtain

$|K_{2i}(\mathcal{O}_{F})|\sim_{p}\left\\{\begin{array}[]{ll}\zeta_{F}\big{(}1-(i+1)\big{)},&\hbox{$1\leq
i<d-1$, $i$ odd}\\\
p^{a(F)}\zeta_{F}\big{(}1-d\big{)},&\hbox{$i=d-1$.}\end{array}\right.$

Therefore, statement (2) of Theorem 6.1 is equivalent to saying that $p$
divides one of the numerator of the following rational numbers

$\zeta_{F}\big{(}1-(i+1)\big{)}\quad(\mbox{$1\leq i<d-1$, $i$ odd}),\quad
p^{a(F)}\zeta_{F}\big{(}1-d\big{)}.$

It thus follows that our theorem is equivalent to [10, Theorem 1] and [13,
Theorem 1].

We proceed with the proof of Theorem 6.1, where we emphasize that our proof
does not make use of $p$-adic $L$-functions. We however should make the
following remark.

######  Remark 6.3.

Although the proof of Theorem 6.1 does not make use of $p$-adic $L$-function,
in order to see that the said theorem is equivalent to Kummer’s criterion, one
requires to be able to relate the special values of the Dedekind zeta function
and the size of the $K$-groups, and such relation is predicted by a conjecture
of Lichtenbaum [18]. At our current knowledge, it would seem that the only way
to study this relation is via the Iwasawa main conjecture, whose formulation
itself will require the $p$-adic $L$-functions.

For the proof of Theorem 6.1, we require the following lemma.

######  Lemma 6.4.

Retain the setting of Theorem 6.1. For every odd integer $i$ such that $1\leq
i\leq d-1$, we have

$r_{p}\big{(}K_{2i}(\mathcal{O}_{F})\big{)}=r_{p}\big{(}\varepsilon_{-i}A_{F(\zeta_{p})}\big{)}.$

###### Proof.

In the proof, we shall write $E=F(\zeta_{p})$. Recall that by Corollary 4.3,
one has

$r_{p}\big{(}K_{2i}(\mathcal{O}_{F})\big{)}=r_{p}\big{(}\varepsilon_{-i}A^{S}_{E}\big{)}+|S_{p}^{(i)}|.$

From the assumption that the primes in
$S_{p}\big{(}F(\zeta_{p}+\zeta_{p}^{-1})\big{)}$ do not split in $E$, we see
that $|\Delta_{v}|$ is even for every $v\in S_{p}$. But since $i$ is odd, this
in turn implies that the set $S_{p}^{(i)}$ is empty. It therefore remains to
show that

$\varepsilon_{-i}A^{S}_{E}=\varepsilon_{-i}A_{E}$

for every odd $i$. Since $d$ is even, this is equivalent to showing that
$\varepsilon_{i}A^{S}_{E}=\varepsilon_{i}A_{E}$ for every odd $i$. Let
$\lambda$ be the natural surjection
$\operatorname{Cl}(\mathcal{O}_{E})\twoheadrightarrow\operatorname{Cl}_{S}(\mathcal{O}_{E})$.
Then $\ker\lambda$ is generated by the class of primes in $S_{p}(E)$. Now
observe that for an odd integer $i$, we have

$\displaystyle\varepsilon_{i}=$
$\displaystyle~{}\frac{1}{d}\sum_{k=0}^{d-1}\omega(g)^{-ik(p-1)/d}\sigma^{-k(p-1)/d}$
$\displaystyle=$
$\displaystyle~{}\frac{1}{d}\sum_{k=0}^{\frac{d}{2}-1}\left(\omega(g)^{ik(p-1)/d}\sigma^{-k(p-1)/d}+\omega(g)^{i(k+\frac{d}{2})(p-1)/d}\sigma^{-(k+\frac{d}{2})(p-1)/d}\right)$
$\displaystyle=$
$\displaystyle~{}\frac{1}{d}\sum_{k=0}^{\frac{d}{2}-1}\omega(g)^{ik(p-1)/d}\sigma^{-k(p-1)/d}\left(1+(-1)^{i}\sigma^{-\frac{d}{2}(p-1)/d}\right)$
$\displaystyle=$
$\displaystyle~{}\frac{1}{d}\sum_{k=0}^{\frac{d}{2}-1}\omega(g)^{ik(p-1)/d}\sigma^{-k(p-1)/d}\left(1-\sigma^{\frac{p-1}{2}}\right),$

where we note that $\sigma^{-\frac{p-1}{2}}=\sigma^{\frac{p-1}{2}}$ is the
complex conjugation and hence a generator of the group
$\operatorname{Gal}(E/F(\zeta_{p}+\zeta_{p}^{-1}))$. Since the primes in
$S_{p}\big{(}F(\zeta_{p}+\zeta_{p}^{-1})\big{)}$ do not split in
$F(\zeta_{p})$, they are invariant under the complex conjugation
$\sigma^{\frac{p-1}{2}}$ and so $\ker\lambda$ is annihilated by
$1-\sigma^{\frac{p-1}{2}}$. This in turn implies that
$(\ker\lambda)[p^{\infty}]$ is annihilated by $\varepsilon_{i}$ for every odd
$i$, which in turn yields $\varepsilon_{i}A^{S}_{E}=\varepsilon_{i}A_{E}$ for
every odd $i$. The proof of the lemma is now complete. ∎

###### Proof of Theorem 6.1.

Let $A$ denote the Sylow $p$-subgroup of the class group of $E:=F(\zeta_{p})$.
Then there is a decomposition

$A=\bigoplus_{i=0}^{d-1}\varepsilon_{i}A,$

where one notes that $\displaystyle\bigoplus_{\begin{subarray}{c}0\leq i\leq
d-1\\\ i~{}\mathrm{even}\end{subarray}}\varepsilon_{i}A$ is the Sylow
$p$-subgroup of the class group of $F(\zeta_{p}+\zeta_{p}^{-1})$. By [10,
Section 4], the class number of $E$ is divisible by $p$ if and only the
relative class number of $E/F(\zeta_{p}+\zeta_{p}^{-1})$ is divisible by $p$.
In view of the above decomposition, this latter divisibility is therefore
equivalent to

$r_{p}\left(\bigoplus_{\begin{subarray}{c}1\leq i\leq d-1\\\
i~{}\mathrm{odd}\end{subarray}}\varepsilon_{i}A\right)\geq 1.$

By Lemma 6.4, this in turn is equivalent to
$r_{p}\big{(}K_{2i}(\mathcal{O}_{F})\big{)}\geq 1$ for some odd $i$ with
$1\leq i\leq d-1$. The proof of the theorem is therefore complete. ∎

## References

  * [1] P. Báyer and J. Neukirch, On values of zeta functions and $l$-adic Euler characteristics. Invent. Math. 50 (1978/79), no. 1, 35-64.
  * [2] S. Bloch; K. Kato, $p$-adic étale cohomology. Inst. Hautes Études Sci. Publ. Math. No. 63 (1986), 107-152.
  * [3] A. Borel, Stable real cohomology of arithmetic groups. Ann. Sci. École Norm. Sup. (4) 7 (1974), 235-272.
  * [4] J. Browkin, On the $p$-rank of the tame kernel of algebraic number fields. J. Reine Angew. Math. 432 (1992), 135-149.
  * [5] X. Y. Cheng, Tame kernels of pure cubic fields. Acta Math. Sin. (Engl. Ser.) 28 (2012), no. 4, 771-780.
  * [6] J. Coates, On $K_{2}$ and some classical conjectures in algebraic number theory. Ann. of Math. (2) 95 (1972), 99-116.
  * [7] W. G. Dwyer; E. Friedlander, Algebraic and etale $K$-theory. Trans. Amer. Math. Soc. 292 (1985), no. 1, 247-280.
  * [8] H, Garland, A finiteness theorem for $K_{2}$ of a number field. Ann. of Math. (2) 94 (1971), 534-548.
  * [9] G. Gras, Remarks on $K_{2}$ of number fields. J. Number Theory 23 (1986), no. 3, 322-335.
  * [10] R. Greenberg, A generalization of Kummer’s criterion. Invent. Math. 21 (1973), 247-254.
  * [11] J.-F. Jaulent, La théorie de Kummer et le $K_{2}$ des corps de nombres. Sém. Théor. Nombres Bordeaux (2) 2 (1990), no. 2, 377-411.
  * [12] F. Keune, On the structure of the $K_{2}$ of the ring of integers in a number field. K-Theory 2 (1989), no. 5, 625-645.
  * [13] M. Kida, Kummer’s criterion for totally real number fields, Tokyo J. Math. 14 (1991), no. 2, 309-317.
  * [14] M. Kolster, Odd torsion in the tame kernel of totally real number fields. Algebraic K-theory: connections with geometry and topology (Lake Louise, AB, 1987), 177–188, NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., 279, Kluwer Acad. Publ., Dordrecht, 1989.
  * [15] M. Kolster, $K_{2}$ of rings of algebraic integers. J. Number Theory 42 (1992), no. 1, 103-122.
  * [16] M. Kolster, $K$-theory and arithmetic. Contemporary developments in algebraic $K$-theory, 191-258, ICTP Lect. Notes, XV, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2004.
  * [17] Y. Li; H. Zhou; F. Deng; X. Wu, The $p$-rank of tame kernels of pure quintic fields. Algebra Colloq. 25 (2018), no. 2, 277-284.
  * [18] S. Lichtenbaum, On the values of zeta and $L$-functions. I. Ann. of Math. (2) 96 (1972), 338-360.
  * [19] J. Milne, Arithmetic Duality Theorems. Second edition. BookSurge, LLC, Charleston, SC, 2006. viii+339 pp.
  * [20] J. Milnor, Algebraic $K$-theory and quadratic forms. Invent. Math. 9 (1969/70), 318-344.
  * [21] J. Neukirch; A. Schmidt; K. Wingberg, Cohomology of Number Fields, 2nd edn., Grundlehren Math. Wiss. 323 (Springer-Verlag, Berlin, 2008).
  * [22] J. Rognes; C. Weibel, Two-primary algebraic $K$-theory of rings of integers in number fields. Appendix A by Manfred Kolster. J. Amer. Math. Soc. 13 (2000), no. 1, 1-54.
  * [23] P. Schneider, Über gewisse Galoiscohomologiegruppen. Math. Z. 168 (1979), no. 2, 181-205.
  * [24] C. Soulé, $K$-théorie des anneaux d’entiers de corps de nombres et cohomologie étale. Invent. Math. 55 (1979), no. 3, 251-295.
  * [25] C. Sun; K. Xu, On the $p$-rank of tame kernel of number fields. J. Number Theory 158 (2016), 244-267.
  * [26] H. Qin, Reflection theorems and the $p$-Sylow subgroup of $K_{2}\mathcal{O}_{F}$ for a number field $F$. J. Pure Appl. Algebra 214 (2010), no. 7, 1181-1192.
  * [27] D. Quillen, Higher algebraic $K$-theory. I. Algebraic $K$-theory, I: Higher $K$-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pp. 85-147. Lecture Notes in Math., Vol. 341, Springer, Berlin 1973.
  * [28] D. Quillen, Finite generation of the groups $K_{i}$ of rings of algebraic integers. Algebraic $K$-theory, I: Higher $K$-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pp. 179-198. Lecture Notes in Math., Vol. 341, Springer, Berlin, 1973.
  * [29] J. Tate, Letter from Tate to Iwasawa on a relation between $K_{2}$ and Galois cohomology. Algebraic $K$-theory, II: “Classical” algebraic $K$-theory and connections with arithmetic (Proc. Conf., Seattle Res. Center, Battelle Memorial Inst., 1972), pp. 524-527. Lecture Notes in Math., Vol. 342, Springer, Berlin, 1973.
  * [30] J. Tate, Relations between $K_{2}$ and Galois cohomology. Invent. Math. 36 (1976), 257-274.
  * [31] V. Voevodsky, On motivic cohomology with $\mathbf{Z}/l$-coefficients. Ann. of Math. (2) 174 (2011), no. 1, 401-438.
  * [32] C. Weibel, The norm residue isomorphism theorem. J. Topol. 2 (2009), no. 2, 346-372.
  * [33] C. Weibel, The $K$-book. An introduction to algebraic $K$-theory. Graduate Studies in Mathematics, 145. American Mathematical Society, Providence, RI, 2013. xii+618 pp.
  * [34] A. Wiles, The Iwasawa conjecture for totally real fields. Ann. of Math. (2) 131 (1990), no. 3, 493-540.
  * [35] X. Wu; Z. Zhao, Reflection theorems and the tame kernel of a number field. Comm. Algebra 42 (2014), no. 7, 2779-2787.
  * [36] H. Zhou, On the $p^{2}$-rank of tame kernels of quadratic fields. Comm. Algebra 35 (2007), no. 9, 2810-2819.
  * [37] H. Zhou, On tame kernels and ideal class groups. Acta Math. Sin. (Engl. Ser.) 23 (2007), no. 10, 1807-1812.
  * [38] H. Zhou, On the odd part of tame kernels of biquadratic number fields. Acta Arith. 144 (2010), no. 3, 275-286.
  * [39] H. Zhou; Z. Liang, Tame kernels of cubic and sextic fields. J. Number Theory 190 (2018), 277-291.