# Testing $\mathrm{Lyman}\,\alpha$ emission line reconstruction routines at
multiple velocities in one system.

David J. Wilson Laboratory for Atmospheric and Space Physics, University of
Colorado at Boulder, 600 UCB, Boulder, CO 80303  Allison Youngblood Goddard
Space Flight Center, Greenbelt, MD 20771  Laboratory for Atmospheric and Space
Physics, University of Colorado at Boulder, 600 UCB, Boulder, CO 80303  Odette
Toloza Departamento de Física, Universidad Técnica Federico Santa María,
Avenida España 1680, Valparaíso, Chile  Jeremy J. Drake Center for
Astrophysics — Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138,
USA  Kevin France Laboratory for Atmospheric and Space Physics, University of
Colorado at Boulder, 600 UCB, Boulder, CO 80303  Cynthia S. Froning McDonald
Observatory, University of Texas at Austin, Austin, TX 78712  Boris T.
Gänsicke Department of Physics, University of Warwick, Coventry CV4 7AL, UK
Seth Redfield Wesleyan University, Department of Astronomy and Van Vleck
Observatory, 96 Foss Hill Dr., Middletown, CT 06459, USA  Brian E. Wood Naval
Research Laboratory, Space Science Division, Washington, DC 20375, USA

###### Abstract

The 1215.67 Å H I $\mathrm{Lyman}\,\alpha$ emission line dominates the
ultraviolet flux of low mass stars, including the majority of known exoplanet
hosts. Unfortunately, strong attenuation by the interstellar medium (ISM)
obscures the line core at most stars, requiring the intrinsic
$\mathrm{Lyman}\,\alpha$ flux to be reconstructed based on fits to the line
wings. We present a test of the widely-used $\mathrm{Lyman}\,\alpha$ emission
line reconstruction code lyapy using phase-resolved, medium-resolution STIS
G140M observations of the close white dwarf-M dwarf binary EG UMa. The Doppler
shifts induced by the binary orbital motion move the $\mathrm{Lyman}\,\alpha$
emission line in and out of the region of strong ISM attenuation.
Reconstructions to each spectrum should produce the same
$\mathrm{Lyman}\,\alpha$ profile regardless of phase, under the well-justified
assumption that there is no intrinsic line variability between observations.
Instead, we find that the reconstructions underestimate the
$\mathrm{Lyman}\,\alpha$ flux by almost a factor of two for the lowest-
velocity, most attenuated spectrum, due to a degeneracy between the intrinsic
$\mathrm{Lyman}\,\alpha$ and ISM profiles. Our results imply that many stellar
$\mathrm{Lyman}\,\alpha$ fluxes derived from G140M spectra reported in the
literature may be underestimated, with potential consequences for, for
example, estimates of extreme-ultraviolet stellar spectra and ultraviolet
inputs into simulations of exoplanet atmospheres.

††facilities: HST(COS,STIS)††software: astropy (Astropy Collaboration, 2018),
stistools555https://stistools.readthedocs.io/en/latest/, scipy (Virtanen et
al., 2020), numpy (Harris et al., 2020), matplotlib (Hunter, 2007), emcee
(Foreman-Mackey et al., 2013)

##  1 Introduction

The H I $\mathrm{Lyman}\,\alpha$ line at 1215.67 Å is a paradox of stellar
observational astronomy: Vitally important for the study of stellar
atmospheres and their effects on orbiting exoplanets, contributing as it does
a large fraction of the ultraviolet flux of low mass stars ($\approx$30–70 %,
France et al. 2013); but extremely difficult to observe, occulted almost
entirely by a combination of absorption by hydrogen in the local interstellar
medium (ISM) and geocoronal airglow emission from the Earth’s atmosphere.
Airglow is very uniform on the angular size scale of stars observed with
astronomical ultraviolet spectrometers, and its effects can generally be
corrected for by subtracting the off-source background spectrum. However, the
ISM is so opaque to $\mathrm{Lyman}\,\alpha$ photons that, near the rest
wavelength of the line in the frame of reference of the intervening ISM, even
emission from the nearest stars is completely absorbed. Nevertheless,
observations of the $\mathrm{Lyman}\,\alpha$ line are the focus of a large
number of papers and observing programs, returning data from hundreds of stars
to date. These observations have been used to, for example, measure stellar
winds (Wood et al., 2021), estimate the extreme ultraviolet flux of stars
(Linsky et al., 2014), characterise the outer heliosphere (Wood et al., 2014),
and model the atmospheres of the TRAPPIST-1 planets (Bourrier et al., 2017;
Wunderlich et al., 2020).

Stellar $\mathrm{Lyman}\,\alpha$ observations rely on the fact that the
$\mathrm{Lyman}\,\alpha$ lines of their targets are broad enough that the
wings of the line are detectable on one or both sides of the region of strong
ISM absorption and/or airglow. The intrinsic profile and flux of the
$\mathrm{Lyman}\,\alpha$ line can then be reconstructed from the wings.
Multiple methods have been developed for the reconstruction, such as using
metal lines to characterise and thus remove the ISM (Wood et al., 2005) and/or
fitting model line profiles to the wings (Bourrier et al., 2015; Youngblood et
al., 2016), with broad agreement between techniques and recipes. However,
statistical uncertainties in the reconstructions, which range from 5 % to 100
% depending on the data quality and whether the unsaturated deuterium
absorption line from the ISM is spectrally resolved, are dominated by
degeneracies between the ISM absorbers and intrinsic stellar profile, as well
as our incomplete knowledge of the intrinsic $\mathrm{Lyman}\,\alpha$ profile
shape for the vast majority of stars. Improving the accuracy of
$\mathrm{Lyman}\,\alpha$ reconstructions is particularly important for
accurately modeling chemistry in exoplanet atmospheres. Small changes of
$\sim$20 % in the reconstructed $\mathrm{Lyman}\,\alpha$ flux can propagate to
$\sim$30 % changes in the O2 and O3 column depths in Earth-like planets
orbiting M dwarfs (Segura et al., 2007). In mini-Neptune atmospheres,
$\mathrm{Lyman}\,\alpha$ is the dominant driver of photochemistry in the
atmospheric layers most likely to be probed by future direct observations
(Miguel et al., 2015).

Testing the absolute accuracy of the $\mathrm{Lyman}\,\alpha$ reconstruction
routines is challenging, as the ground truth of occulted
$\mathrm{Lyman}\,\alpha$ profiles cannot be obtained. In rare cases the
$\mathrm{Lyman}\,\alpha$ line can be fully observed for stars with
sufficiently high radial velocities to shift the line out of the airglow and
the deepest ISM absorption, the best example being Kapteyn’s star at 245 km
s-1 (Guinan et al., 2016; Schneider et al., 2019; Youngblood et al., 2022;
Peacock et al., 2022). Unfortunately, high-velocity stars cannot also be
observed with the line occulted. What is required is a star with a radial
velocity that changes by many 10s of km s-1 over time, allowing observations
to be taken at low velocities when the $\mathrm{Lyman}\,\alpha$ line is
occulted, and the reconstruction based on those data compared with high-
velocity, unobscured $\mathrm{Lyman}\,\alpha$ observations of the same star.
Such conditions exist in detached Post Common Envelope Binaries (PCEBs),
specifically those containing a white dwarf with a main-sequence companion.

EG UMa is a PCEB comprising an M4-type dwarf star and a white dwarf in a 16
hour orbit(Lanning, 1982; Bleach et al., 2000) at a distance of $\approx 28$
pc. The white dwarf is unusually cool among the sample of well-studied PCEBs,
with an effective temperature of only around 13000 K (Sion et al., 1984), such
that the ultraviolet emission from the system is not completely dominated by
the white dwarf and shows a detectable contribution from the M dwarf. The
short orbital period provides the large range of radial velocities (up to
$\approx 150$ km s-1) that allows the system to be observed at multiple levels
of $\mathrm{Lyman}\,\alpha$ occultation, providing a comprehensive test of
$\mathrm{Lyman}\,\alpha$ reconstruction routines. This paper presents the
results of this experiment using phase-resolved Hubble Space Telescope (HST)
spectroscopy of EG UMa.

##  2 Observations

Table 1: Summary of observations. Dataset numbers are given for retrieval from MAST (https://archive.stsci.edu/hst/). Date | Instrument | Grating | Central | Start Time (UT) | Exposure Time (s) | Dataset | Orbital Phase
---|---|---|---|---|---|---|---
|  |  | Wavelength (Å) |  |  |  |
2017-12-05 | COS | G130M | 1291 | 17:30:29 | 1969 | LDLC05010 | –
2021-04-02 | STIS | G140M | 1222 | 06:39:15 | 1358 | OEHUA4010 | 0.53
2021-04-02 | STIS | G140M | 1222 | 09:51:27 | 1358 | OEHUA1010 | 0.73
2021-04-02 | STIS | G140M | 1222 | 14:35:35 | 1358 | OEHUA2010 | 0.03
2021-04-26 | STIS | G140M | 1222 | 19:46:59 | 1358 | OEHUB3010 | 0.3

###  2.1 HST/COS

Table 1 provides a log of the observations of EG UMa discussed in this
section. All data used in this paper can be found in MAST:
http://dx.doi.org/10.17909/k8h7-s556 (catalog 10.17909/k8h7-s556). EG UMa was
observed with the Cosmic Origins Spectrograph (COS, Green et al., 2012)
onboard HST as part of program ID 15189. A single observation was obtained on
2017 December 05 with an exposure time of 1969 s, using the Primary Science
Aperture and the G130M grating with a central wavelength of 1291 Å. The
spectrum was automatically extracted using the standard CalCOS tools. The
spectrum is shown in Figure 1.

###  2.2 HST/STIS

Four additional HST spectra of EG UMa were obtained using the Space Telescope
Imaging Spectrograph (STIS, Woodgate et al., 1998) on 2021 April 02 (three
spectra) and 2021 April 26 (one spectrum)111This observation was initially
attempted on 2021 March 30, but a technical issue caused a complete data loss.
We note it here to avoid confusion for readers wishing to retrieve the data.
The failed observation is archived on MAST as dataset OEHUA3010. under program
ID 16449. Each spectrum had an exposure time of 1358 s and was obtained using
the G140M grating with a central wavelength of 1222 Å and the 52X0.1 arcsecond
slit. The spectra were timed to obtain data at the 0.0, 0.25, 0.5, and 0.75
phases of the binary orbit based on the ephemeris of Bleach et al. (2002),
with an allowable phase error of $\pm 0.05$. The observations were further
scheduled such that the radial velocity of the Earth relative to the predicted
ISM velocity ($1.45\pm 1.37$ km s-1, Redfield & Linsky 2008) along the line of
sight to EG UMa was no more than $\pm 20$ km s-1, placing the bright
geocoronal $\mathrm{Lyman}\,\alpha$ airglow line in the deepest region of ISM
absorption. The airglow contribution was further reduced by the use of the
narrow 0.1 arcsecond slit. The spectral trace was automatically located and
extracted by the standard CalSTIS pipeline. An example STIS spectrum is shown
in the top panel of Figure 2. We used the stistools splittag routine to
extract time-series spectra from each observation, and found no evidence for
variability of the $\mathrm{Lyman}\,\alpha$ line during any of the visits.

We evaluated the flux calibration of the STIS spectra by comparison with the
COS spectrum. Three of the four spectra showed good agreement at all
wavelengths, apart from the expected Doppler shift. Spectrum OEHUA2010 showed
significant, unphysical departures from the COS spectrum at wavelengths
$<1210$ Å, including a region of negative flux and a region roughly 1.5 times
higher than the COS flux. We attribute the poor extraction to “FUV glow”, an
irregular region of dark current on the STIS MAMA detector that grows over
time after the detector is powered on after an SAA passage222https://hst-
docs.stsci.edu/stisihb/chapter-7-feasibility-and-detector-
performance/7-5-mama-operation-and-feasibility-
considerations#id-7.5MAMAOperationandFeasibilityConsiderations-7.5.2MAMADarks.
OEHUA2010 was the third of three EG UMa observations on the same day, and the
irregular dark current had increased such that the standard background regions
($\pm 300$ pixels from the spectral trace) were no longer representative of
the background around the spectral trace. We re-extracted OEHUA2010 with the
stistools x1d routine, changing the background regions to $\pm 50$ pixels from
the trace, which brought the spectrum into much better agreement with the COS
spectrum, although with a lower S/N ratio at the shorter wavelengths compared
to the other STIS spectra.

##  3 COS spectral analysis and experiment design

Figure 1: Far ultraviolet spectrum of EG UMa. The COS spectrum is shown in
blue, with a dashed line indicating the region affected by geocoronal
$\mathrm{Lyman}\,\alpha$ airglow. Prominent emission (from the M dwarf
atmosphere) and absorption (stellar wind accretion onto the white dwarf)
features are labelled. The white dwarf model spectrum is shown in orange. The
STIS spectrum shows the highest-velocity and mostly unobscured
$\mathrm{Lyman}\,\alpha$ line at Phase 0.73, which exceeds the axis limit by
factor $\approx 2$. The inset shows the region around the C II 1335 Å lines,
simultaneously fit with an astropy model to measure the instantaneous radial
velocity of the M dwarf (emission features), ISM (weak absorption features)
and the white dwarf (strong absorption features).

Figure 1 shows the far-ultraviolet spectrum of EG UMa. The spectrum is
dominated by the extremely broad $\mathrm{Lyman}\,\alpha$ absorption feature
in the white dwarf photosphere, along with the satellite feature around 1400 Å
(Allard et al., 1994; Koester et al., 2014). Although not strictly correct, we
refer to this as the white dwarf continuum hereafter to avoid confusion with
the M dwarf $\mathrm{Lyman}\,\alpha$ emission line. The spectrum was fit with
synthetic models generated by the latest version of the code described in
Koester (2010), using the parallax measurements from Gaia DR2 (Gaia
Collaboration et al., 2018) to constrain the distance. As the results are of
marginal relevance to this work we do not detail the fitting process here, but
point the reader to Wilson et al. (2021a) for a full description of the
methods used to fit the data from program 15189. We find an effective
temperature $T_{\mathrm{eff}}$$=12599\pm 13$ K and $\log g$$=7.83\pm 0.01$
(statistical uncertainties only) and infer a white dwarf mass of $\approx
0.51$$\mathrm{M}_{\odot}$. Our $T_{\mathrm{eff}}$ is substantially lower than
the $\approx 18000$ K found in recent studies (e.g., Gianninas et al., 2011;
Limoges et al., 2015), but those results were based on optical data that
feature significant flux contributions from the companion, which can seriously
affect the standard analysis of the Balmer absorption lines of the white
dwarf. Sion et al. (1984) found $T_{\mathrm{eff}}$$=13000\pm 500$ K based on
data from the International Ultraviolet Explorer, consistent with our result.

In addition to the continuum, multiple emission and absorption features are
detected. The emission features originate from the outer atmosphere of the M
dwarf, with the same collection of lines seen in ultraviolet spectra of
isolated mid-M dwarfs (see for e.g., Loyd et al., 2016). The absorption
features originate in the white dwarf photosphere and are produced by
accretion of the M dwarf stellar wind onto the white dwarf (Debes, 2006). Of
particular interest is the C II doublet around 1335 Å which shows both
emission and absorption features, as well as weak ISM absorption (inset in
Figure 1). We fit the C II lines using the astropy (Astropy Collaboration,
2018) modelling functions, using a 2nd order polynomial for the continuum and
six Gaussian profiles for the lines, with the line separation and widths fixed
between each pair of features. Although only one ISM feature is reliably
detected, we find a line-of-sight ISM velocity of $3.7\pm 2.8$ km s-1,
consistent with the $1.45\pm 1.37$ km s-1 velocity predicted by the ISM maps
from Redfield & Linsky (2008) and significantly less than the COS wavelength
solution accuracy of $\pm 7.5$ km s-1. The O I lines at 1300 Å also show the
same emission-ISM-absorption configuration, but as these lines are blended
with nearby Si lines and geocoronal O I airglow we did not attempt to fit
them.

The combination of strong detected emission lines and high radial velocity
amplitude suggests that this is an ideal system to test the
$\mathrm{Lyman}\,\alpha$ reconstruction routines by observing the
$\mathrm{Lyman}\,\alpha$ line at different velocities and therefore different
levels of obscuration by ISM absorption. Unfortunately the large aperture of
COS means that the stellar $\mathrm{Lyman}\,\alpha$ line is completely swamped
by bright geocoronal emission over the full radial velocity range (dashed line
in Figure 1). The experiment was therefore performed using STIS with a narrow
(0.1 arcsecond) slit that greatly reduced the airglow contribution.

##  4 STIS results

Figure 2: Top panel: Example STIS G140M spectrum of EG UMa (dataset OEHUA2010)
with major emission features marked. Note the increased scatter noise at
wavelengths $\lesssim 1214$ Å due to FUV glow. Bottom panels: The
$\mathrm{Lyman}\,\alpha$ emission line in all four spectra, in the observer
rest frame (left) and velocity-shifted to the M dwarf rest frame (right). The
dashed purple line in the left panel shows the center-of-mass radial velocity
of the binary system.

As expected, the phase-resolved STIS observations returned four distinctly
different $\mathrm{Lyman}\,\alpha$ profiles. Figure 2 shows the region around
the $\mathrm{Lyman}\,\alpha$ line, velocity adjusted to the observer frame
(left) and M dwarf rest frame (right). The observations clearly demonstrate
the effects of ISM absorption on the $\mathrm{Lyman}\,\alpha$ line, with the
line being much fainter at low velocities than higher. The STIS G140M spectra
also cover the Si III 1206 Å and N V 1240 Å emission lines, which are clearly
detected in all four spectra.

Given the allowed uncertainty in the phase targeting and the cycle-count
errors induced by the twenty-year gap between these observations and those of
Bleach et al. (2002), the phases of each observation are not quite at the 0,
0.25, 0.5 and 0.75 positions. Our recalculation of the correct phase positions
is described in Section 5.2, but to avoid confusion we use the corrected
phases (0.03, 0.3, 0.53 and 0.73) in figures and text throughout the paper.

##  5 Lyman Alpha reconstructions

###  5.1 Line model and reconstruction routine

The observed $\mathrm{Lyman}\,\alpha$ spectrum is a combination of four
components: Geocoronal airglow from the Earth; Absorption from the ISM;
Absorption from the white dwarf photosphere; and finally the feature that we
are aiming to characterise, emission from the chromosphere of the M dwarf. Our
observations were timed such that the radial velocity of the Earth coincided
with the expected radial velocity of the ISM in the direction of EG UMa,
ensuring that the airglow emission fell in the deepest region of ISM
absorption and would not overlap with the measurable signal from EG UMa.
Combined with the standard pipeline background subtraction, we find that
airglow makes a negligible contribution to the spectra and can be ignored. The
flux from the white dwarf in the region around the $\mathrm{Lyman}\,\alpha$
core is much smaller than that from even the wings of the M dwarf emission
line and can also be neglected. This leaves the ISM absorption and intrinsic
stellar emission to be fitted.

We reconstructed the $\mathrm{Lyman}\,\alpha$ line for each spectrum with the
latest version of the publicly-available lyapy333lyapy is actively developed
at https://github.com/allisony/lyapy. The version used in this paper is
avaiable at https://doi.org/10.5281/zenodo.6949067. routines (Youngblood et
al., 2016, 2021). The intrinsic emission from the M dwarf is treated as a
Voigt profile (McLean et al., 1994) modelled using the Astropy Voigt-1D
function. Youngblood et al. (2022) demonstrated that significant self-reversal
is present at the cores of $\mathrm{Lyman}\,\alpha$ lines observed for high-
velocity M dwarfs, so we optionally include a second, absorbing Voigt profile
at the line core. The broad ($>$100 km s-1) ISM contribution is a combination
of absorption from H I and D I, with rest wavelengths of 1215.67 Å and 1215.34
Å respectively, which we also treat as a pair of Voigt profiles. The STIS
G140M grating has insufficient resolution to resolve the D I line, so the
ratio of the column density of the absorption features is fixed at
$D/H=1.5\times 10^{-5}$ (Linsky et al., 2006) and the Doppler parameter $b$,
which determines the width of the features, fixed as $b_{HI}=11.5$ km s-1,
$b_{DI}=b_{HI}/\sqrt{2}$ (Wood et al., 2004; Redfield & Linsky, 2004). The
observed spectra can then be modelled by multiplying the normalised ISM
profile with the $\mathrm{Lyman}\,\alpha$ line and convolving the resulting
spectrum with the STIS G130M line spread
function444https://www.stsci.edu/hst/instrumentation/stis/performance/spectral-
resolution. The parameters to be fit are therefore the velocity, amplitude and
Gaussian and Lorentzian widths of the $\mathrm{Lyman}\,\alpha$ emission line
$V_{Ly\alpha}$, $A$, $FWHM_{G}$ and $FWHM_{L}$, a dimensionless self-reversal
parameter $p$ (if included), along with the hydrogen column density N(H I) and
ISM velocity $V_{\mathrm{ISM}}$. The intrinsic $\mathrm{Lyman}\,\alpha$ flux
($F_{Ly\alpha}$), which is the key result for most astrophysical applications,
is calculated by numerical integration over the Voigt profile. Assuming that
the intrinsic $\mathrm{Lyman}\,\alpha$ emission profile has not changed
between observations, fits to all four spectra should return the same result
for all parameters except for $V_{Ly\alpha}$. $V_{Ly\alpha}$ should in turn be
consistent with the velocities measured from the Si III 1206.5 Å and N V
1238.8, 1242.8 Å emission lines in the same spectrum. These lines were also
fit with Voigt profiles to measure their fluxes and radial velocities. For the
N V doublet we fixed the line separation, and fixed the ratio of the line
strength to two, the ratio of their respective oscillator strengths.

As the velocity of the M dwarf changes over each exposure, the emission lines
may be smeared out the changing Doppler shift. We used the phase positions to
calculate the change in velocity over each exposure. The change in velocity is
$\leq 18$ km s-1 for all spectra, smaller than the instrumental resolution of
$\approx 27$ km s-1. Nevertheless, we added a Doppler broadening effect to our
fitting routines for our initial $\mathrm{Lyman}\,\alpha$ reconstructions. The
differences between reconstructions with and without broadening included were
negligible. As including broadening is computationally expensive, we did not
include it in further fits. The line is also subject to rotational broadening
of $\approx 30$ km s-1 (Bleach et al., 2002), but as it is constant with phase
we did not include it as a specific parameter in the mode.

We fit the profiles using a Markov-Chain Monte Carlo (MCMC) method as
implemented by emcee (Foreman-Mackey et al., 2013). emcee maximizes the sum of
the logarithm of our parameters’ prior probabilities and the logarithm of a
likelihood function that measures the goodness of fit of the model to the
data. We assume uniform priors for all parameters (Youngblood et al., 2016)
and a Gaussian likelihood function. $\log$ N(H I) was forced to be $>17$, and
if self-reversal was included then $p$ was forced to be $\geq 1$. We used 100
walkers, ran for 50 autocorrelation times, and removed a burn-in period.

In addition to fitting each spectrum individually, we also performed a
simultaneous fit to all four spectra. In this case the intrinsic
$\mathrm{Lyman}\,\alpha$ and ISM profiles were forced to be the same for all
four spectra, with only $V_{Ly\alpha}$ varying between reconstructions. Whilst
fitting to the $\mathrm{Lyman}\,\alpha$ line at different velocities is
impossible for single stars, we can use this joint fit to inform our
discussion of the individual fits, as well as take advantage of this unique
dataset to get the best possible reconstruction of the EG UMa
$\mathrm{Lyman}\,\alpha$ line.

###  5.2 Reconstruction Results

Figure 3: Fits to the $\mathrm{Lyman}\,\alpha$ emission line in all four G140M
spectra, ordered by orbital phase. The observed spectrum is shown in blue with
errorbars on every third point. The intrinsic $\mathrm{Lyman}\,\alpha$ profile
is shown in green and the ISM transmission in purple, both of which are
convolved to the instrumental resolution. The product of these profiles is the
final fit to the data (orange). Shaded areas behind each profile show the 1
$\sigma$ uncertainties of the model fits. The bottom panel shows the
residuals, calculated as (data-model)/data uncertainty. Figure 4: As Figure 3,
but fitting all four spectra simultaneously with only the radial velocity
allowed to vary between reconstructions. Figure 5: Summary of the
reconstruction results: a. The intrinsic flux of the best-fit
$\mathrm{Lyman}\,\alpha$ line for each spectrum, comparing fits with and
without self reversal (SR); b. The radial velocities of the emission lines; c.
The integrated fluxes of the Si III and N V emission lines, where N V is the
combined flux of the doublet; d. The ISM hydrogen column density; e. The Voigt
widths of the best-fit $\mathrm{Lyman}\,\alpha$ line; and f. the radial
velocity of the ISM. The dashed lines show the mean value of each quantity,
except for panel b. where the radial velocity curve of the M dwarf is shown.
In all panels the red points at Phase $=1.0$ show the results for the
simultaneous fit to all four spectra, which are not included in the means. All
results are for fits including self reversal except where noted in panel A.
Small phase offsets have been applied where necessary to avoid overlapping
points.  Figure 6: Left: Intrinsic $\mathrm{Lyman}\,\alpha$ line profiles
reconstructed from the individual spectra, along with the simultaneous fit to
all four, velocity-shifted to the rest frame. Results from fits not including
self-reversal are shown as dashed lines. Right: The same for the ISM profile.
In both panels the profiles have been convolved with the STIS G140M line
spread function.

The reconstructed $\mathrm{Lyman}\,\alpha$ profiles for each spectrum are show
in Figure 3 (individual fits) and Figure 4 (simultaneous fit). The key
parameters are summarised in Figure 5 and Table 2. Figure 6 shows an overlay
of the intrinsic $\mathrm{Lyman}\,\alpha$ and ISM profiles.

For the individual fits, we find that each reconstruction successfully
recreates the observed spectrum, with the exception of the possibly spurious
feature at $\approx 100$ km s-1 at Phase 0.73. The velocities of the emission
lines are also consistent within each spectrum. We fit the radial velocity
curve from Bleach et al. (2000) to the measured velocities, bounding the
orbital velocity amplitude, net velocity and period within their respective
uncertainties and allowing the phase to vary freely (Figure 5, panel b). This
refit velocity curve was used to calculate the correct phase position of each
spectrum, with the corrected phases used throughout this paper. In all cases
we find only small differences in fits with and without self reversal, which
we discuss in more detail below, but the discussion hereafter will refer to
the results from fits including self reversal unless otherwise stated.

Contrary to our expectations, we find different reconstructed
$\mathrm{Lyman}\,\alpha$ emission profiles and intrinsic fluxes for all four
spectra, with the peak of the intrinsic profile approximately following the
peak of the observed flux. No correlation is seen between the
$\mathrm{Lyman}\,\alpha$ flux and the Si III and N V fluxes: In fact, the
emission line flux is highest when the $\mathrm{Lyman}\,\alpha$ flux is
lowest. The reconstructions at Phases 0.3, 0.53 and 0.73 are broadly similar,
with intrinsic fluxes within 1–2$\sigma$. However at Phase 0.03, where none of
the line core is observed, the intrinsic flux is only $\approx$ 50–70 % that
found for the other spectra, with a 15 $\sigma$ difference between the highest
and lowest reconstructed fluxes. The $FWHM_{G}$ is also larger for Phase 0.03.
We find that the profile reconstructed from all four spectra simultaneously
returns a good fit to all four observed profiles, although the RMS residuals
are $\approx 5$–20 % higher than for the individual fits. Comparing the
profiles in Figure 6, we find that the intrinsic $\mathrm{Lyman}\,\alpha$
profile is similar to that found when fitting only the least-attenuated
observed profile (Phase 0.73) but that a deeper and broader ISM profile is
required for a good fit to all four spectra.

The fitted properties of the ISM N(H I) and $V_{\mathrm{ISM}}$ are within 1–2
$\sigma$ for all four spectra, with the mean $V_{\mathrm{ISM}}$ (1.6 km s-1)
in agreement with the $3.7\pm 2.8$ km s-1 measurement from the COS spectrum.
However, the spread in the best-fit values of N(H I) is nearly an order of
magnitude, with the lower values likely unphysical (the column density to
Alpha Centauri is N(H I) = 17.6, Wood et al. 2001). Despite this, swapping the
fitted ISM profiles between spectra and calculating the predicted flux returns
reasonable matches to the data in all cases. The most likely reason for the
wide allowed range of N(H I) is the low resolution of the G140M grating, and
in particular the inability to resolve the D I line.

##  6 Discussion

###  6.1 Why are the fits different?

The differences in reconstructed $\mathrm{Lyman}\,\alpha$ flux between the
four spectra may have troubling implications for existing
$\mathrm{Lyman}\,\alpha$ fits to dozens of stars in the literature. However,
we must first rule out that the different results are due to genuine
variability and/or consequences of binarity.

The EG UMa white dwarf may influence the observations in two ways:
Contribution from the (velocity-shifting) underlying spectrum and heating of
the facing side of the M dwarf. We see no evidence for contributions from the
white dwarf spectrum to our fits, as the contribution from the
$\mathrm{Lyman}\,\alpha$ emission line wings is over an order of magnitude
brighter than the modelled white dwarf spectrum at the same wavelengths, and
none of the modelled parameters in Figure 5 vary in phase with the white dwarf
motion (that is, in anti-phase to the M dwarf velocity). The irradiation from
the white dwarf is also insufficient to power strong variations in emission
line strength. Using equation 7 from Rebassa-Mansergas et al. (2013) we find
that the ratio of the flux generated by irradiation to the inherent flux from
the M dwarf is $\approx 0.5$ %, in keeping with the few % variations seen in
optical photometic data (e.g., Bleach et al., 2000). Additionally, if
irradiation were causing the flux variations we would expect to see the
highest emission line flux at phase 0.53 when the full day side of the M dwarf
is visible, but the highest fluxes are observed at Phases 0.03 (N V, Si III)
and 0.73 ($\mathrm{Lyman}\,\alpha$).

All M dwarfs demonstrate some level of variability in the ultraviolet (Loyd et
al., 2018; France et al., 2020), including flares that can produce factor
10–100 increases in emission line fluxes for short time periods (Froning et
al., 2019). The flux of the N V and Si III lines are stronger by a factor
$\sim 2$ at phase 0.03, indicating that EG UMa may have been undergoing a
flare during that exposure. However this does not explain the discrepancy in
$\mathrm{Lyman}\,\alpha$ flux, as the response of the $\mathrm{Lyman}\,\alpha$
line to flares is generally small (Loyd et al., 2018), and the reconstructed
$\mathrm{Lyman}\,\alpha$ flux at Phase 0.03 is lower than at other phases, not
higher. Explaining the results as genuine variability would also be a strong
appeal to coincidence, with the variably exactly mimicking a constant-strength
line moving in and out of a region of high ISM attenuation. Finally, the fact
that the simultaneous fit does return a good fit to all four spectra indicates
that there is at least one $\mathrm{Lyman}\,\alpha$ profile that is consistent
with all observations. In summary, there is neither firm direct nor
circumstantial evidence that the intrinsic $\mathrm{Lyman}\,\alpha$ line
profile changed during our observations.

As described in Section 2.2, the spectrum at Phase 0.03 required a custom
extraction. To check that our results were not affected by the re-extraction,
we performed an additional $\mathrm{Lyman}\,\alpha$ reconstruction based on
the original automatic extraction. We found only an $\approx$ 10 % change in
the reconstructed intrinsic $\mathrm{Lyman}\,\alpha$ flux between the
automatic and custom extractions, insufficient to explain the discrepancy with
the other observations.

We therefore conclude that the fault is not in our star but in our
reconstructions. The differences between the reconstructions to Phases 0.3,
0.53 and 0.73 are small, so can be discussed together as the high-velocity
fits. These fits are clearly incompatible with Phase 0.03. Shifting the
intrinsic profile from any of the high-velocity phases to the velocity of
Phase 0.03 and attenuating by the ISM from either fit results in a model
spectrum that over predicts the observed flux by a factor $\approx 2$.
Conversely, the intrinsic flux reconstructed from the Phase 0.03 data under
predicts the observed flux at the higher velocity observations.

However, the success of the simultaneous fit shows that there is at least one
reconstruction that fits all four phases, and that lyapy is able to find it
when sufficiently constrained. Comparing the individual fits to the
simultaneous fit suggests an explanation for the different reconstructions:
Degeneracy between the strength of the line core and the ISM absorption. The
larger intrinsic flux in the simultaneous fit than in the individual fit to
Phase 0.03 is offset by a wider ISM profile. With none of the line core
observed and the D I line unresolved, there is no information to distinguish
between a weak $\mathrm{Lyman}\,\alpha$ core with low ISM attenuation, or
strong $\mathrm{Lyman}\,\alpha$ with proportionally strong attenuation. In the
high-velocity fits more of the line core is detected, better constraining the
balance between the intrinsic profile and ISM with information that is
unavailable at low velocities.

###  6.2 Implications for observations at single stars.

The majority of $\mathrm{Lyman}\,\alpha$ reconstructions are carried out
single stars (or individual members of wide binaries) with low radial
velocities, based on STIS G140M/L spectra. The spectra of their
$\mathrm{Lyman}\,\alpha$ lines appear similar to the observation at Phase 0.03
presented here, with the emission line core completely occulted by the ISM
(see for e.g. France et al., 2013; Youngblood et al., 2016; Bourrier et al.,
2018; Youngblood et al., 2021; Wilson et al., 2021b). The published
uncertainties for these reconstructions generally fall in the range 5–30 %, so
the factor two difference found here represents a significant increase in
uncertainty. Inaccurate $\mathrm{Lyman}\,\alpha$ measurements propagate into,
for example, incorrect integrated far-ultraviolet fluxes and/or inaccurate
estimates of the ultraviolet input into the atmospheres of orbiting
exoplanets. On the other hand, a factor-two change in $\mathrm{Lyman}\,\alpha$
flux is small in comparison to order-of-magnitude level uncertainties in
attempts to determine the flux of the far-ultraviolet continuum (Teal et al.,
2022).

If the uncertainly in $\mathrm{Lyman}\,\alpha$ flux is found to have a
significant effect on the outcomes of exoplanet atmosphere observations,
future $\mathrm{Lyman}\,\alpha$ reconstructions of low velocity data may need
to incorporate more detailed priors. For example, the attenuation may be
estimated from maps of nearby ISM clouds (e.g Redfield & Linsky, 2008).
Current formula for estimating the $\mathrm{Lyman}\,\alpha$ strength from
other emission lines (Youngblood et al., 2017; Melbourne et al., 2020) are too
imprecise to improve the priors here, but these relationships may improve with
future observations. Although it does not appear to have been a factor in this
case, the radial velocity of the $\mathrm{Lyman}\,\alpha$ line could also be
constrained via (for example) measurements of the nearby S III and N V
emission lines. Ideally, higher resolution STIS E140M spectra shoud be
obtained, which can resolve the D I and tightly constrain the ISM profile, but
the lower throughput of E140M makes this impractical for many targets. High-
resolution observations of the Mg II H&K lines around 2800 Å obtained
contemporaneously with $\mathrm{Lyman}\,\alpha$ observations, can provide
additional constraints on the ISM line velocity and column depth (see for e.g.
Carleo et al., 2021).

###  6.3 Self reversal

Self-reversal occurs in optically-thick emission lines where the line center
traces hot, high-altitude, low-density gas that departs significantly from
Local Thermodynamic Equilibrium. As a result, thermal emission is less
efficient in the line core than the wings, producing to a line profile with a
reversed core. Youngblood et al. (2022) performed $\mathrm{Lyman}\,\alpha$
reconstructions for a selection of high radial velocity stars, where the self-
reversal of the $\mathrm{Lyman}\,\alpha$ core could be detected. Among their
results was that including self-reversal was essential to recover the true
$\mathrm{Lyman}\,\alpha$ flux, with reconstructions without self reversal
overestimating the $\mathrm{Lyman}\,\alpha$ flux by factors of 40–170 %. In
contrast to that, our reconstructions to EG UMa find little difference whether
self-reversal is included or not, both in the integrated flux (Figure 5 a.)
and profile (Figure 6). Spatially resolved observations of active regions on
the Sun have consistently shown decreased self reversal in emission line
profiles (e.g. Schmit et al., 2015) in comparison with inactive regions, so
the lower self-reversal at EG UMa could be due to it being more active than
the stars characterised by Youngblood et al. (2022). Alternatively,
observational effects could be to blame. Although we found that orbital
smearing had no effect on the overall reconstructions, the combination of
smearing, rotational broadening and the relatively low resolution G140M
grating may have complicated the retrieval of any contribution from self-
reversal in the line cores.

###  6.4 Potential future targets

A caveat to our results is that we only perform this experiment for a single
system, and similar observations of other PCEBs may produce different
outcomes. The ideal criteria for such observations include: a white dwarf with
a wide $\mathrm{Lyman}\,\alpha$ line (i.e., relatively cool) so that the M
dwarf emission line can be reliably isolated; A radial velocity amplitude
$\gtrsim 100$ km s-1 so that the line peak moves through a range of ISM
attenuation; Detectable M dwarf emission lines so that an estimate of the
$\mathrm{Lyman}\,\alpha$ strength can be made before investing valuable HST
time; and a strong enough estimated $\mathrm{Lyman}\,\alpha$ flux to be
detected in short, single orbit exposures with the STIS G140M grating, or
ideally the higher resolution E140M grating.

Extensive STIS E140M data have already been obtained for the white dwarf-K
dwarf binary V471 Tau. Unfortunately the white dwarf makes a considerable,
variable contribution to the region around the $\mathrm{Lyman}\,\alpha$ core
and extensive post-pipeline data processing is required to isolate the
emission line (see for e.g. Sion et al., 2012; Wood et al., 2005). We
performed an initial analysis of the V471 Tau spectra and found that the
change in the detected $\mathrm{Lyman}\,\alpha$ emission line morphology is
small in comparison to EG UMa. Further processing the data to the required
level is beyond the scope of this paper, although we recommend this experiment
as a secondary science goal in any future analysis of the V471 Tau dataset.
Another PCEB, HZ 9, was also observed with E140M as part of the same program,
but none of the four spectra were obtained at low radial velocities.

Hernandez et al. (2022) present a STIS G140L spectrum of the white dwarf-G
dwarf binary TYC 110-755-1. Despite a velocity amplitude of only $\approx 50$
km s-1 the $\mathrm{Lyman}\,\alpha$ emission line is clearly detected and
distinguishable from the $\approx 16600$ K white dwarf, as are multiple other
emission lines. Further STIS spectroscopy of this star at different phases
would allow an excellent test of the $\mathrm{Lyman}\,\alpha$ reconstruction
routines at a much earlier spectral type to EG UMa.

In addition to EG UMa, Bleach et al. (2000) characterised its near-twin PG
1026+002 (UZ Sex), an M4 spectral type star orbiting a 17000 K white dwarf
with a $\approx 200$ km s-1 radial velocity amplitude. COS spectroscopy of PG
1026+002 (dataset LDLC03010, Wilson et al. in prep.) shows a similarly broad
$\mathrm{Lyman}\,\alpha$ absorption line to EG UMa and, although no
ultraviolet emission lines are detected, Ca II H&K and H $\alpha$ emission
lines are present in optical spectra (Napiwotzki et al., 2020) allowing an
estimate of the $\mathrm{Lyman}\,\alpha$ strength. PG 1026+002 is therefore an
ideal target to repeat this experiment, comparing EG UMa with a potentially
less active example of a similar spectral type.

##  7 Conclusion

By observing the $\mathrm{Lyman}\,\alpha$ emission of the binary star EG UMa
at multiple velocities, we have found that the widely used lyapy
reconstruction routines return different results as a function of how much of
the $\mathrm{Lyman}\,\alpha$ core is detected, with a difference of factor
$\approx 2$ between the strongest and weakest reconstruction. We find that
this is unlikely to be due to intrinsic variation from the star, both as there
is no plausible physical mechanism to provide such variation, and as a
simultaneous fit to all observations returns a single consistent result. We
suggest that the issue is a degeneracy between the strength of the ISM
attenuation and $\mathrm{Lyman}\,\alpha$ flux, especially at low velocities
where none of the line core is detected. $\mathrm{Lyman}\,\alpha$
reconstructions using medium-resolution spectra of single, low-velocity stars
may therefore need additional prior constraints on the
$\mathrm{Lyman}\,\alpha$ and ISM profiles to provide a precise
$\mathrm{Lyman}\,\alpha$ flux measurement.

This research is based on observations made with the NASA/ESA Hubble Space
Telescope obtained from the Space Telescope Science Institute, which is
operated by the Association of Universities for Research in Astronomy, Inc.,
under NASA contract NAS 5–26555. These observations are associated with
programs 15189 and 16449. Support for program 16449 was provided by NASA
through a grant from the Space Telescope Science Institute, which is operated
by the Association of Universities for Research in Astronomy, Inc., under NASA
contract NAS 5-03127.BTG was supported by grant ST/T000406/1 from the Science
and Technology Facilities Council (STFC). This project has received funding
from the European Research Council (ERC) under the European Union’s Horizon
2020 research and innovation programme (Grant agreement No. 101020057). We
thank the HST operations team for their work ensuring the accurate phasing of
the STIS observations, and Daniel Welty at STScI for assistance with the FUV
glow issue.

##  Appendix A Table of results

Table 2 shows the results of our $\mathrm{Lyman}\,\alpha$ reconstructions for
the four STIS spectra.

|  | With Self Reversal | Without Self Reversal |  |  |  |  |  |
---|---|---|---|---|---|---|---|---|---
Dataset | Phase | $F_{Ly\alpha}$ | $V_{Ly\alpha}$ | $p$ | $F_{Ly\alpha}$ | $V_{Ly\alpha}$ | F${}_{Si\,II}$ | $V_{Si\,II}$ | $F_{N\,V}$ | $V_{N\,V}$ | N(H I) (cm-2) | $V_{\mathrm{ISM}}$
OEHUA2010 | 0.03 | 1.06${}^{+0.05}_{-0.04}$ | -17.8${}^{+3.4}_{-3.4}$ | 1.15${}^{+0.21}_{-0.11}$ | 1.16${}^{+0.09}_{-0.07}$ | -19.0${}^{+3.3}_{-3.2}$ | 0.025${}^{+0.006}_{-0.006}$ | -18.6${}^{+7.7}_{-8.1}$ | 0.060${}^{+0.006}_{--0.029}$ | -4.6${}^{+3.0}_{-2.8}$ | 17.2${}^{+0.2}_{-0.1}$ | 0.7${}^{+2.6}_{-1.7}$
OEHUB3010 | 0.3 | 1.71${}^{+0.12}_{-0.14}$ | 74.3${}^{+7.2}_{-5.2}$ | 1.04${}^{+0.07}_{-0.03}$ | 1.71${}^{+0.12}_{-0.12}$ | 78.7${}^{+5.0}_{-4.4}$ | 0.013${}^{+0.003}_{-0.004}$ | 76.1${}^{+7.8}_{-7.9}$ | 0.024${}^{+0.004}_{--0.007}$ | 94.0${}^{+6.6}_{-7.1}$ | 17.9${}^{+0.1}_{-0.2}$ | 0.6${}^{+6.4}_{-2.1}$
OEHUA4010 | 0.53 | 1.54${}^{+0.11}_{-0.10}$ | -72.1${}^{+4.2}_{-4.8}$ | 1.10${}^{+0.19}_{-0.08}$ | 1.65${}^{+0.13}_{-0.11}$ | -70.8${}^{+4.0}_{-3.9}$ | 0.010${}^{+0.003}_{-0.003}$ | -70.4${}^{+11.6}_{-11.1}$ | 0.022${}^{+0.004}_{--0.008}$ | -51.7${}^{+5.3}_{-4.8}$ | 17.7${}^{+0.1}_{-0.1}$ | -6.2${}^{+2.0}_{-2.6}$
OEHUA1010 | 0.73 | 1.83${}^{+0.05}_{-0.04}$ | -177.1${}^{+2.3}_{-1.9}$ | 1.06${}^{+0.10}_{-0.05}$ | 1.97${}^{+0.15}_{-0.09}$ | -171.1${}^{+8.2}_{-4.4}$ | 0.016${}^{+0.004}_{-0.004}$ | -151.0${}^{+6.4}_{-5.7}$ | 0.015${}^{+0.004}_{-0.005}$ | -167.9${}^{+11.3}_{-11.0}$ | 17.3${}^{+0.2}_{-0.2}$ | 4.9${}^{+5.4}_{-6.2}$
All | – | 1.98${}^{+0.04}_{-0.04}$ | – | 1.01${}^{+0.02}_{-0.01}$ | 2.02${}^{+0.04}_{-0.04}$ | – | 0.042${}^{+0.003}_{-0.004}$ | – | 0.175${}^{+0.005}_{-0.005}$ | – | 18.0${}^{+0.0}_{-0.0}$ | -0.2${}^{+1.0}_{-0.7}$
Table 2: Results of the STIS analysis by dataset. All line fluxes have units
of 10-13 erg s-1 cm-2 Å-1, velocities are in km s-1.

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# On the index of power compositional polynomials

Sumandeep Kaur  ,  Surender Kumar  and  László Remete  Department of
Mathematics, Panjab University, Chandigarh  Department of Mathematics, Indian
Institute of Technology (IIT), Bhilai  Institute of Mathematics, University of
Debrecen, Hungary and ELKH-DE Equations, Functions, Curves and their
Applications Research Group<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

The index of a monic irreducible polynomial $f(x)\in\mathbb{Z}[x]$ having a
root $\theta$ is the index $[\mathbb{Z}_{K}:\mathbb{Z}[\theta]]$, where
$\mathbb{Z}_{K}$ is the ring of algebraic integers of the number field
$K=\mathbb{Q}(\theta)$. If $[\mathbb{Z}_{K}:\mathbb{Z}[\theta]]=1$, then
$f(x)$ is monogenic. In this paper, we give necessary and sufficient
conditions for a monic irreducible power compositional polynomial $f(x^{k})$
belonging to $\mathbb{Z}[x]$, to be monogenic. As an application of our
results, for a polynomial $f(x)=x^{d}+A\cdot h(x)\in\mathbb{Z}[x],$ with
$d>1,\deg h(x)<d$ and $|h(0)|=1$, we prove that for each positive integer $k$
with $\operatorname{rad}(k)\mid\operatorname{rad}(A)$, the power compositional
polynomial $f(x^{k})$ is monogenic if and only if $f(x)$ is monogenic,
provided that $f(x^{k})$ is irreducible. At the end of the paper, we give
infinite families of polynomials as examples.

###### Key words and phrases:

discriminant, monogenity of polynomials, power compositional polynomials.

###### 2010 Mathematics Subject Classification:

11R04, 11R09.

The second author is grateful to the University Grants Commision, New Delhi
for providing financial support in the form of Junior Research Fellowship
through Ref No.1129/(CSIR-NET JUNE 2019). The research of the third named
author was supported in part by the Eötvös Loránd Research Network (ELKH) and
by the Univerity of Debrecen Scientific Research Bridging Fund (DETKA)

##  1\. Introduction and statements of the results

Let $f(x)\in\mathbb{Z}[x]$ be a monic irreducible polynomial with a root
$\theta$, and set $K=\mathbb{Q}(\theta)$. If $\mathbb{Z}_{K}$ denotes the ring
of algebraic integers of $K$, then the index of $f(x)$ is the index
$[\mathbb{Z}_{K}:\mathbb{Z}[\theta]]$. It is well known that the index of
$f(x)$ is always finite. We say that $f(x)$ is monogenic, if the index of
$f(x)$ is equal to $1$.

Let $k$ be a positive integer and $q$ be a prime divisor of $k$. Suppose
$f(x)\in\mathbb{Z}[x]$ is a monic polynomial. If $f(x^{k})$ is irreducible,
then the roots $\alpha$ and $\beta$ of $f(x^{q})$ and $f(x^{k})$ respectively,
give rise to a tower of fields as shown below.

$\mathbb{Q}$$\mathbb{Q}(\alpha)$$\mathbb{Q}(\beta)$

Note that $\beta^{\frac{k}{q}}=\alpha$. In this paper, our main motive is to
discuss the relation between the indices $[\mathbb{Z}_{L}:\mathbb{Z}[\beta]]$
and $[\mathbb{Z}_{K}:\mathbb{Z}[\alpha]]$, where $L=\mathbb{Q}(\beta)$ and
$K=\mathbb{Q}(\alpha)$, i.e., the relation between the index of $f(x^{q})$ and
the index of $f(x^{k}),$ where $q\mid k$. Moreover, we give necessary and
sufficient conditions for these indices to be equal to $1$.
It is well known, that if $f(x)$ is the minimal polynomial of an algebraic
integer $\theta$ over $\mathbb{Q}$, then the discriminant $D(f)$ of $f(x)$ and
the discriminant $\Delta(K)$ of $K=\mathbb{Q}(\theta)$ are related by the
formula

$D(f)=[\mathbb{Z}_{K}:\mathbb{Z}[\theta]]^{2}\cdot\Delta(K).$ (1.1)

Clearly, if $D(f)$ is squarefree, then $f(x)$ is monogenic. But the converse
is not always true. Several classes of monogenic polynomials with non
squarefree discriminants are known and their properties have been studied (cf.
[3] \- [4], [6] \- [15], [19]). L. Jones in [13] studied the monogenity of
power compositional Shanks polynomials. In [14], L. Jones constructed infinite
collections of monic Eisenstein polynomials $f(x)\in\mathbb{Z}[x]$ such that
the power compositional polynomials $f(x^{d^{n}})$ are monogenic for all
integers $n\geq 0$ and any integer $d>1$, where $f(x)$ is Eisenstein with
respect to every prime divisor of $d$. In this paper, we study the monogenity
of a monic irreducible power compositional polynomial; i.e., a monic
irreducible polynomial of the form $f(x^{k})\in\mathbb{Z}[x]$.
Throughout the paper, the irreducibility of a monic polynomial belonging to
$\mathbb{Z}[x]$ is meant over $\mathbb{Q}$, unless stated otherwise. Note,
that for a positive integer $k$, the irreducibility of a monic polynomial
$f(x^{k})$ implies the irreduciblity of $f(x)$. For a positive integer $z,$
the radical of $z$, denoted by $\operatorname{rad}(z)$, is defined as the
largest squarefree integer dividing $z$.
Next, we state our main theorems.

######  Theorem 1.1.

Let $f(x)$ be a monic polynomial with integer coefficients. Let $k\geq 2$ be
an integer such that $f(x^{k})$ is irreducible. Then $f(x^{k})$ is monogenic
if and only if

  1. (1)

$f(x)$ is monogenic,

  2. (2)

$p$ does not divide the index of $f(x^{p})$ for all primes $p\mid k$ and

  3. (3)

$f(0)$ is squarefree.

######  Remark 1.2.

Note, that for a monic polynomial $f(x)\in\mathbb{Z}[x]$, if the irreducible
polynomial $f(x^{k})$ is monogenic, then by using Lemma 2.6 it is easy to
conclude that $f(x^{t})$ is also monogenic for each divisor $t$ of $k$.

In the proof of Theorem 1.1 we will see that if $f(x)$ is monogenic, then only
the prime divisors of $k$ and $f(0)$ can divide the index of $f(x^{k})$, so we
obtain the following corollary to be used in the proof of Theorem 1.8.

######  Corollary 1.3.

Let $f(x)$ be a monic polynomial having integer coefficients. Let $k$ be a
positive integer such that $f(x^{k})$ is irreducible. Suppose, that

  1. (1)

$f(x)$ is monogenic and

  2. (2)

$p$ does not divide the index of $f(x^{p})$ for any prime $p\mid k$.

Then every prime divisor of the index of $f(x^{k})$ will divide $f(0)$.

In view of the above corollary, if $|f(0)|=1$ or the divisors of $f(0)$ do not
divide the index of $f(x^{k})$, then (1) and (2) together are equivalent to
the monogenity of $f(x^{k})$. This happens, for example, in the case of
polynomials that are Eisenstein with respect to any prime divisor of $f(0)$,
see Lemma 2.12. These polynomials can be given of the form $f(x)=x^{d}+A\cdot
h(x)$, where $|h(x)|=1$ and $A$ is squarefree. Adding a further condition, the
monogenity of $f(x)$ will be equivalent to the monogenity of $f(x^{k})$.

######  Corollary 1.4.

Let $f(x)\in\mathbb{Z}[x]$ be a polynomial of the form $f(x)=x^{d}+A\cdot
h(x)$, where $d>1,|h(0)|=1$ and $\deg h(x)<d$. Let $k$ be a positive integer
such that $f(x^{k})$ is irreducible. If $\operatorname{rad}(k)$ divides
$\operatorname{rad}(A)$, then $f(x)$ is monogenic if and only if $f(x^{k})$ is
monogenic.

Note, that Theorem 1.1 implies that if $f(x^{p^{u}})$ is monogenic, then
$f(x)$ is monogenic and $f(0)$ is squarefree. In the following theorem we
prove, that its converse is true with an additional requirement. More
precisely, we give sufficient conditions for the monogenity of a monic
irreducible prime power compositional polynomial.

######  Theorem 1.5.

Let $u$ be a positive integer, $p$ a prime number and $f(x)\in\mathbb{Z}[x]$ a
monic polynomial. Suppose that $f(x^{p^{u}})$ is irreducible. Then the
following statements hold:

  1. (1)

If $f(x)$ is monogenic and $f(0)$ is squarefree, then the index of
$f(x^{p^{u}})$ is equal to $p^{s}$, with some non-negative integer $s$.

  2. (2)

$p$ does not divide the index of $f(x^{p^{u}})$ if and only if
$\frac{f(x^{p})-f(x)^{p}}{p}$ is coprime to $f(x)$ modulo $p$.

######  Remark 1.6.

Note, that if $u=1$ in the above theorem, then the condition (2) of Theorem
1.1 is equivalent to the condition that $\frac{f(x^{p})-f(x)^{p}}{p}$ is
coprime to $f(x)$ modulo $p$, for every prime $p\mid k$.

Combining the Corollary 1.4 and Theorem 1.5 we get the statement below, which
also follows from Lemma 3.1 of L. Jones [14].

######  Corollary 1.7.

Let $p$ be a prime number. Suppose, that $f(x)\in\mathbb{Z}[x]$ is a monic and
Eisenstein polynomial with respect to $p$. If $f(0)$ is squarefree, then
$f(x)$ is monogenic if and only if $f(x^{p^{u}})$ is monogenic for any
positive integer $u$.

Taking Remark 1.6 into account, when $f(x)=x^{d}+A\cdot h(x)\in\mathbb{Z}[x]$,
with $|h(0)|=1$ and $\deg h(x)<d$, we can provide necessary and sufficient
conditions for the monogenity of the irreducible power compositional
polynomial $f(x^{k})$. As an application of this theorem, we constructed an
infinite family of polynomials for a specific choice of $h(x)$, given in the
last section.

######  Theorem 1.8.

Let $f(x)\in\mathbb{Z}[x]$ be a polynomial of the form $f(x)=x^{d}+A\cdot
h(x)$, where $d>1,|h(0)|=1$ and $\deg h(x)<d$. Let $k$ be a positive integer
such that $f(x^{k})$ is irreducible. Then $f(x^{k})$ is monogenic if and only
if

  1. (1)

$f(x)$ is monogenic, and

  2. (2)

$\frac{A\cdot h(x^{p})+(-A\cdot h(x))^{p}}{p}$ is coprime to $f(x)$ modulo
$p$, for any prime $p\mid k$.

##  2\. Preliminary results

In what follows, for a prime $p$ and a polynomial
$g(x)\in\mathbb{Z}[x],~{}\overline{g}(x)$ will denote the polynomial obtained
by reducing each coefficient of $g(x)$ modulo $p$. For a subset
$S\subset\mathbb{Z}[x]$, let $\langle S\rangle$ be the ideal generated by $S$.
If $S=\\{s_{1},s_{2},\ldots,s_{k}\\}$ is finite, then

$\langle S\rangle=\\{s_{1}a_{1}+s_{2}a_{2}+\ldots+s_{k}a_{k}\mid
a_{i}\in\mathbb{Z}[x],~{}1\leq i\leq k\\}.$

To investigate the monogenity of $f(x)$, we will follow the approach of K.
Uchida, so we recall Lemma 2.A and Theorem 2.B proved in [20].

Lemma 2.A. An ideal $\mathfrak{M}$ of $\mathbb{Z}[x]$ containing a monic
polynomial is maximal if and only if $\mathfrak{M}=\langle p,g(x)\rangle$ for
some prime number $p$ and a monic polynomial $g(x)$ belonging to
$\mathbb{Z}[x]$ which is irreducible modulo $p$.

Theorem 2.B. Let $K=\mathbb{Q}(\theta)$ with $\theta$ in $\mathbb{Z}_{K}$
having minimal polynomial $f(x)$ over $\mathbb{Q}$, then
$\mathbb{Z}_{K}=\mathbb{Z}[\theta]$ if and only if $f(x)$ does not belong to
$\mathfrak{M}^{2}$ for any maximal ideal $\mathfrak{M}$ of the polynomial ring
$\mathbb{Z}[x]$.

In particular, a prime $p$ divides the index
$[\mathbb{Z}_{K}:\mathbb{Z}[\theta]]$ if and only if $f(x)$ belongs to the
square of a maximal ideal of the form $\mathfrak{M}=\langle p,g(x)\rangle$,
where $g(x)\in\mathbb{Z}[x]$ is a monic polynomial which is irreducible modulo
$p$. We can split the condition $f(x)\in\langle p,g(x)\rangle^{2}$ into two
parts.

######  Lemma 2.1.

Let $p$ be a prime and $g(x)\in\mathbb{Z}[x]$ be a monic polynomial which is
irreducible modulo $p$. Then $\langle p,g(x)\rangle^{2}=\langle
p^{2},g(x)\rangle\cap\langle p,g^{2}(x)\rangle.$

###### Proof.

Clearly, $\langle p,g(x)\rangle^{2}\subseteq\langle
p^{2},g(x)\rangle\cap\langle p,g^{2}(x)\rangle$. To obtain the other direction
of the containment, suppose that the polynomial $f(x)$ belongs to both
$\langle p^{2},g(x)\rangle$ and $\langle p,g^{2}(x)\rangle$, i.e., there exist
$a(x),b(x),c(x),d(x)$ belonging to $\mathbb{Z}[x]$, such that

$p^{2}\cdot a(x)+g(x)\cdot b(x)=f(x)=p\cdot c(x)+g^{2}(x)\cdot d(x).$

Since $g(x)$ is irreducible modulo $p$, it is irreducible in $\mathbb{Z}[x]$.
So $\langle g(x)\rangle$ is a prime ideal of $\mathbb{Z}[x]$. By the above
equation, $p\cdot(c(x)-p\cdot a(x))$ is a multiple of $g(x)$, i.e., it belongs
to the prime ideal $\langle g(x)\rangle$. Obviously, $p$ does not belong to
$\langle g(x)\rangle$, thus $c(x)-p\cdot a(x)\in\langle g(x)\rangle$ and
therefore $c(x)\in\langle p,g(x)\rangle$. This implies that $p\cdot
c(x)\in\langle p^{2},p\cdot g(x)\rangle$, and from there $f(x)=p\cdot
c(x)+g^{2}(x)\cdot d(x)\in\langle p^{2},p\cdot g(x),g^{2}(x)\rangle$. Hence
$f(x)\in\langle p,g(x)\rangle^{2}.$ ∎

If $f(x)\in\langle p,g^{2}(x)\rangle$, then $\overline{g}(x)$ is a multiple
factor of $\overline{f}(x)$. On the other hand, $f(x)\in\langle
p^{2},g(x)\rangle$ implies that the remainder of $f(x)$ divided by $g(x)$ is a
multiple of $p^{2}$. Therefore, the results of Uchida [20] say that $p$
divides the index of $f(x)$ if and only if there exists a monic polynomial
$g(x)\in\mathbb{Z}[x]$, which is a multiple irreducible factor of $f(x)$
modulo $p$, such that the remainder of $f(x)$ divided by $g(x)$ is a multiple
of $p^{2}$. This also follows from the results of Ore [17].

######  Corollary 2.2.

Let $f(x)$ and $g(x)$ belonging to $\mathbb{Z}[x]$ be monic polynomials and
let $p$ be a prime number such that $g(x)$ is irreducible modulo $p$. Suppose
that $f(x)\in\langle p,g^{2}(x)\rangle$, then

$f(x)\in\langle p,g(x)\rangle^{2}\iff f(x)\in\langle p^{2},g(x)\rangle.$

Considering a prime divisor $p$ of $f(0)$, it is easy to see that $x$ is a
factor of $f(x)$ modulo $p$ and it is a multiple factor of $f(x^{\ell})$
modulo $p$ if $\ell>1$. So, by using the above corollary, we obtain the
following result.

######  Proposition 2.3.

Let $f(x)\in\mathbb{Z}[x]$ be a monic polynomial and $p$ be a prime, such that
$p^{2}\mid f(0)$. Then $p$ divides the index of $f(x^{\ell})$ for any positive
integer $\ell>1$.

###### Proof.

Since $p^{2}$ divides $f(0)$, we can see that $x$ is a multiple factor of
$f(x^{\ell})$ modulo $p$ for any integer $\ell>1$. Therefore
$f(x^{\ell})\in\langle p,x^{2}\rangle$. Moreover, the remainder of
$f(x^{\ell})$ divided by $x$ is $f(0)$, which is a multiple of $p^{2}$, thus
$f(x^{\ell})\in\langle p^{2},x\rangle$. By Corollary 2.2 and Theorem 2.B, $p$
divides the index of $f(x^{\ell})$. ∎

In the following theorem, we show that if $\bar{g}(x)$ is a multiple
irreducible factor of $\bar{f}(x)$, then the remainder of $f(x)$ divided by
$g(x)$ is a multiple of $p^{2}$ if and only if the remainder of $f(x^{p})$
divided by $g(x)$ is a multiple of $p^{2}$. This observation is important for
the proof of our main results and it is of independent interest as well.

######  Theorem 2.4.

Let $f(x)$ and $g(x)$ belonging to $\mathbb{Z}[x]$ be monic polynomials and
let $p$ be a prime number such that $g(x)$ is irreducible modulo $p$. Suppose
that $f(x)\in\langle p,g^{2}(x)\rangle$. Then

$f(x)\in\langle p,g(x)\rangle^{2}\iff f(x^{p})\in\langle p,g(x)\rangle^{2}.$

###### Proof.

Write $f(x)=g^{2}(x)\cdot q(x)+p\cdot r(x)$ for some $q(x),r(x)$ belonging to
$\mathbb{Z}[x]$. Let $r(x)=g(x)\cdot s(x)+t(x)$, where $\deg t(x)<\deg g(x)$.
Then

$f(x)=g^{2}(x)\cdot q(x)+p\cdot g(x)\cdot s(x)+p\cdot t(x).$ (2.1)

Since $g^{2}(x)\in\langle p,g(x)\rangle^{2}$ and $p\cdot g(x)\in\langle
p,g(x)\rangle^{2}$, we get that

$f(x)\in\langle p,g(x)\rangle^{2}\iff p\cdot t(x)\in\langle
p,g(x)\rangle^{2}.$ (2.2)

Substituting $x^{p}$ in place of $x$ in (2.1),

$f(x^{p})=g^{2}(x^{p})\cdot q(x^{p})+p\cdot g(x^{p})\cdot s(x^{p})+p\cdot
t(x^{p}).$ (2.3)

Using the fact, that $p\mid(g(x^{p})-g(x)^{p})$ we get $g(x^{p})\in\langle
p,g(x)\rangle$. This implies that both $g^{2}(x^{p})$ and $p\cdot g(x^{p})$
belong to the ideal $\langle p,g(x)\rangle^{2}$. Thus, $f(x^{p})\in\langle
p,g(x)\rangle^{2}$ if and only if $p\cdot t(x^{p})\in\langle
p,g(x)\rangle^{2}$. Since $p\mid(t(x)^{p}-t(x^{p}))$ for any polynomial
$t(x)\in\mathbb{Z}[x]$, then

$p\cdot t(x^{p})\in\langle p,g(x)\rangle^{2}\iff p\cdot t(x)^{p}\in\langle
p,g(x)\rangle^{2}.$ (2.4)

Thus, from (2.3) and (2.4),

$f(x^{p})\in\langle p,g(x)\rangle^{2}~{}\iff p\cdot t(x)^{p}\in\langle
p,g(x)\rangle^{2}.$ (2.5)

In view of (2.2) and (2.5), to prove the statement it is enough to show that

$p\cdot t(x)\in\langle p,g(x)\rangle^{2}\text{ if and only if }p\cdot
t(x)^{p}\in\langle p,g(x)\rangle^{2}.$

Clearly, $p\cdot t(x)\in\langle p,g(x)\rangle^{2}$ implies that $p\cdot
t(x)^{p}\in\langle p,g(x)\rangle^{2}$. Assuming that $p\cdot
t(x)^{p}\in\langle p,g(x)\rangle^{2}$, we can write

$p\cdot t(x)^{p}=p^{2}\cdot a(x)+p\cdot g(x)\cdot b(x)+g^{2}(x)\cdot c(x),$

for some $a(x),b(x),c(x)\in\mathbb{Z}[x].$ Therefore, $g^{2}(x)\cdot c(x)$
belongs to the prime ideal $\langle p\rangle$ of $\mathbb{Z}[x]$. Since $g(x)$
is monic and irreducible modulo $p$, then $c(x)\in\langle p\rangle$ and
$t(x)^{p}\in\langle p,g(x)\rangle$. Equivalently, $t(x)\in\langle
p,g(x)\rangle$ and therefore $p\cdot t(x)\in\langle p,g(x)\rangle^{2}$. This
completes the proof. ∎

######  Corollary 2.5.

Let $u$ be a positive integer. For a monic polynomial $f(x)\in\mathbb{Z}[x]$,
a prime $p$ divides the index of $f(x^{p})$ if and only if $p$ divides the
index of $f(x^{p^{u}})$, provided that $f(x^{p^{u}})$ is an irreducible
polynomial.

###### Proof.

Let $g(x)\in\mathbb{Z}[x]$ be a monic polynomial such that $\overline{g}(x)$
is an irreducible factor of $f(x)$ modulo $p$. As $f(x^{p})\equiv f(x)^{p}\mod
p$, $g(x)^{p}$ is a factor of $f(x^{p})$ modulo $p$, i.e., $f(x^{p})\in\langle
p,g^{2}(x)\rangle$. By Theorem 2.4,

$f(x^{p})\in\langle p,g(x)\rangle^{2}\iff f(x^{p^{2}})\in\langle
p,g(x)\rangle^{2}.$

Applying induction on $u$, we get

$f(x^{p})\in\langle p,g(x)\rangle^{2}\iff f(x^{p^{u}})\in\langle
p,g(x)\rangle^{2}.$

Keeping Theorem 2.B in mind, $p$ divides the index of $f(x^{p})$ if and only
if $p$ divides the index of $f(x^{p^{u}})$. ∎

######  Lemma 2.6.

Let $f(x)\in\mathbb{Z}[x]$ be a monic polynomial and $\ell$ a positive integer
such that $f(x^{\ell})$ is irreducible. If a prime $p$ divides the index of
$f(x)$, then $p$ divides the index of $f(x^{\ell})$.

###### Proof.

If $p$ divides the index of $f(x)$, then by Theorem 2.B, there exists a monic
polynomial $g(x)\in\mathbb{Z}[x]$, such that $g(x)$ is irreducible modulo $p$
and $f(x)\in\langle p,g(x)\rangle^{2}$. Therefore, $f(x^{\ell})\in\langle
p,g(x^{\ell})\rangle^{2}$. Let $h(x)$ be an irreducible factor of
$g(x^{\ell})$ modulo $p$. Then $g(x^{\ell})\in\langle p,h(x)\rangle$, so
$\langle p,g(x^{\ell})\rangle\subset\langle p,h(x)\rangle$. This implies, that
$f(x^{\ell})$ belongs to the square of the maximal ideal $\langle
p,h(x)\rangle$. Hence, by Theorem 2.B, $p$ divides the index of $f(x^{\ell})$.
∎

To investigate all of the possible primes that may divide the index of
$f(x^{\ell})$, we need to find its discriminant. We shall use the following
lemma, which describes the properties of the resultant of two polynomials.

######  Lemma 2.7.

Let $f(x)=a_{m}x^{m}+a_{m-1}x^{m-1}+\cdots+a_{0}$ and
$g(x)=b_{n}x^{n}+b_{n-1}x^{n-1}+\cdots+b_{0}$ be two polynomials with integer
coefficients of degrees $m$ and $n$, and let $\lambda_{1},\cdots,\lambda_{m}$
and $\mu_{1}\cdots,\mu_{n}$ be the roots of $f(x)$ and $g(x)$, respectively.
Then

  1. (1)

$R(f,g)=a_{m}^{n}b_{n}^{m}\displaystyle\prod_{\begin{subarray}{c}1\leq i\leq
m\\\ 1\leq j\leq n\end{subarray}}(\lambda_{i}-\mu_{j})$,

  2. (2)

$R(f,g)=a_{m}^{n}\prod\limits_{i=1}^{m}g(\lambda_{i})=(-1)^{mn}b_{n}^{m}\prod\limits_{j=1}^{n}f(\mu_{j})$,

  3. (3)

$R(f,gh)=R(f,g)R(f,h)$, for any $h(x)\in\mathbb{Z}[x]$,

  4. (4)

$R(f,ag)=a^{m}R(f,g)$, for any $a\in\mathbb{Z}$.

Note, that the discriminant of a monic polynomial $f(x)\in\mathbb{Z}[x]$ of
degree $n$ and the resultant $R(f,f^{\prime})$ are related by the following
formula.

$D(f)=(-1)^{\frac{n(n-1)}{2}}R(f,f^{\prime}),$ (2.6)

where $f^{\prime}$ is the derivative of $f.$

######  Lemma 2.8.

Let $f(x)\in\mathbb{Z}[x]$ be a monic polynomial of degree $d$, and $\ell$ a
positive integer. Then the discriminant of $f(x^{\ell})$ is

$D(f(x^{\ell}))=\pm D(f(x))^{\ell}\cdot\ell^{d\ell}\cdot f(0)^{\ell-1}.$

###### Proof.

Using (2.6), we get

$D(f(x^{\ell}))=(-1)^{\frac{d\ell(d\ell-1)}{2}}\cdot R(f(x^{\ell}),\ell
x^{\ell-1}f^{\prime}(x^{\ell})).$

Applying (3) and (4) of Lemma 2.7,

$D(f(x^{\ell}))=(-1)^{\frac{d\ell(d\ell-1)}{2}}\cdot\ell^{d\ell}\cdot
R(f(x^{\ell}),x^{\ell-1})\cdot R(f(x^{\ell}),f^{\prime}(x^{\ell})).$

Using (2) of Lemma 2.7,

$\displaystyle D(f(x^{\ell}))=$
$\displaystyle(-1)^{\frac{d\ell(d\ell-1)}{2}+d\ell+\ell-1}\cdot\ell^{d\ell}\cdot\prod_{i=1}^{\ell-1}f(0)\cdot
R(f(x^{\ell}),f^{\prime}(x^{\ell})),$ $\displaystyle=$
$\displaystyle(-1)^{\frac{d\ell(d\ell+1)}{2}+\ell-1}\cdot\ell^{d\ell}\cdot
f(0)^{\ell-1}\cdot R(f(x^{\ell}),f^{\prime}(x^{\ell})),$ $\displaystyle=$
$\displaystyle(-1)^{\frac{d\ell(d\ell+1)}{2}+\ell-1}\cdot\ell^{d\ell}\cdot
f(0)^{\ell-1}\cdot\prod_{i=1}^{d\ell}f^{\prime}(\lambda_{i}^{\ell}),$

where $f(\lambda_{i}^{\ell})=0$, for $1\leq i\leq d\ell.$ Note, that if
$\mu_{1},\cdots,\mu_{d}$ are roots of $f(x),$ then we can assume that
$\lambda_{i+j-1}^{\ell}=\mu_{j}$ for $1\leq i\leq\ell$ and $1\leq j\leq d.$
Therefore, (1) and (2) of Lemma 2.7 imply that

$\displaystyle D(f(x^{\ell}))=$
$\displaystyle(-1)^{\frac{d\ell(d\ell+1)}{2}+\ell-1}\cdot\ell^{d\ell}\cdot
f(0)^{\ell-1}\cdot\left(\prod_{i=1}^{d}f^{\prime}(\mu_{i})\right)^{\ell}.$

Hence, the result follows from (2) of Lemma 2.7. ∎

The following lemma is proved in [16, Proposition 4.15], and it suggests that
if $f(x)$ is monogenic, then the discriminant of the field generated by a root
of $f(x^{\ell})$ cannot be too small.

######  Lemma 2.9.

Let $K$ and $L$ be number fields with $K\subset L$. Then

$\Delta(K)^{[L:K]}\mid\Delta(L).$

The next proposition is essential for the proof of our main theorems. It
completely describes the possible prime divisors of the index of
$f(x^{\ell})$.

######  Proposition 2.10.

Let $f(x)\in\mathbb{Z}[x]$ be a monic polynomial of degree $d$, $p$ a prime
and $\ell$ an integer greater than $1$. If $p\nmid\ell$ and $f(x^{\ell})$ is
irreducible, then $p$ divides the index of $f(x^{\ell})$ if and only if either

  1. (1)

$p$ divides the index of $f(x)$, or

  2. (2)

$p^{2}\mid f(0)$.

###### Proof.

By Lemma 2.6 and Proposition 2.3, both (1) and (2) imply that $p$ divides the
index of $f(x^{\ell})$. To prove the converse, we assume that $p$ does not
divide the index of $f(x)$ and $p^{2}\nmid f(0)$. We split the proof into two
cases.
Case 1. $p\nmid f(0)$. Equation (1.1) shows, that

$[\mathbb{Z}_{K}:\mathbb{Z}[\theta]]^{2}=\frac{D(f(x))}{\Delta(K)},$

where $\Delta(K)$ is the discriminant of the number field $K$ generated by a
root $\theta$ of $f(x)$. Let $L$ be a number field generated by a root of
$f(x^{\ell})$. From Lemma 2.8,

$D(f(x^{\ell}))=\pm D(f(x))^{\ell}\cdot\ell^{d\ell}\cdot f(0)^{\ell-1}.$

Using Lemma 2.9, we get

$\frac{D(f(x^{\ell}))}{\Delta(L)}\mid\frac{D(f(x))^{\ell}}{\Delta(K)^{\ell}}\cdot\ell^{d\ell}\cdot
f(0)^{\ell-1},$

where $\Delta(L)$ is the discriminant of the number field $L$. Since
$p\nmid\frac{D(f(x))}{\Delta(K)}$ and $p\nmid\ell\cdot f(0)$, then $p$ does
not divide the index of $f(x^{\ell})$.
Case 2. $p\mid f(0)$. In view of Theorem 2.B, we have to show that
$f(x^{\ell})$ does not belong to $\langle p,h(x)\rangle^{2}$, where
$h(x)\in\mathbb{Z}[x]$ is monic and irreducible modulo $p$. By Lemma 2.1, it
is enough to show that $f(x^{\ell})$ does not belong to either $\langle
p,h^{2}(x)\rangle$ or $\langle p^{2},h(x)\rangle$. If $h(x)=x$, then
$f(x^{\ell})-f(0)\in\langle x\rangle\subset\langle p^{2},x\rangle$. But
$f(0)\not\in\langle p^{2},x\rangle$, so $f(x^{\ell})\not\in\langle
p^{2},x\rangle$. First, we show that $f(x^{\ell})$ has a multiple irreducible
factor modulo $p$ different from $x$ if and only if $f(x)$ has a multiple
irreducible factor modulo $p$ different from $x$.

Let $g(x)$ be a monic irreducible polynomial modulo $p$, then
$\gcd(g(x),g^{\prime}(x))=1$. Since the derivative of $g(x^{\ell})$ is $\ell
x^{\ell-1}g^{\prime}(x^{\ell})$ and $p\nmid\ell$, $g(x^{\ell})$ is separable
if and only if $g(x)\neq x$. Thus, $\gcd(g(x^{\ell}),g^{\prime}(x^{\ell}))=1$.
We can conclude, that whenever a monic irreducible polynomial $g(x)\neq x$ is
a single factor of $f(x)$ modulo $p$, then all irreducible factors of
$g(x^{\ell})$ are single factors of $f(x^{\ell})$ modulo $p$. Furthermore, if
$g_{1}(x)$ and $g_{2}(x)$ are distinct irreducible factors of $f(x)$ modulo
$p$, then $\gcd(g_{1}(x),g_{2}(x))=1$ and so
$\gcd(g_{1}(x^{\ell}),g_{2}(x^{\ell}))=1$. Thus, $h(x)\neq x$ is a multiple
irreducible factor of $f(x^{\ell})$ modulo $p$ if and only if $f(x)$ has a
multiple irreducible factor $g(x)$ modulo $p$ such that
$\overline{h}(x)\mid\overline{g}(x^{\ell})$.

Let $g(x)\in\mathbb{Z}[x]$ be a monic polynomial which is a multiple
irreducible factor of $f(x)$ modulo $p$. Then $f(x)\in\langle
p,g^{2}(x)\rangle$. By Corollary 2.2 and Theorem 2.B, with the hypothesis that
$p$ does not divide the index of $f(x)$, we get $f(x)\not\in\langle
p^{2},g(x)\rangle$. Let us write $f(x)=g(x)\cdot c(x)+r(x)$, where $\deg
r(x)<\deg g(x)$ and $c(x),r(x)\in\mathbb{Z}[x]$. Clearly, $r(x)\in\langle
p\rangle$ and $\overline{g}(x)\mid\overline{c}(x)$. Set $r(x)=p\cdot s(x)$,
for some $s(x)\in\mathbb{Z}[x]$. Then, for certain polynomials
$a(x),b(x)\in\mathbb{Z}[x]$, we can write $f(x)=g^{2}(x)\cdot a(x)+p\cdot
g(x)\cdot b(x)+p\cdot s(x)$. Consequently,

$f(x^{\ell})=g^{2}(x^{\ell})\cdot a(x^{\ell})+p\cdot g(x^{\ell})\cdot
b(x^{\ell})+p\cdot s(x^{\ell}).$ (2.7)

As $g(x)$ is irreducible modulo $p$ and $\deg s(x)=\deg r(x)<\deg
g(x)=\deg\overline{g}(x)$, then $\gcd(\overline{g}(x),\overline{s}(x))=1$.
Thus, there exists a polynomial $t(x)\in\mathbb{Z}[x]$ such that $s(x)\cdot
t(x)\in 1+\langle p,g(x)\rangle$ and so $s(x^{\ell})\cdot t(x^{\ell})\in
1+\langle p,g(x^{\ell})\rangle.$ Let $h(x)\in\mathbb{Z}[x]$ be a monic
polynomial which is an irreducible factor of $g(x^{\ell})$ modulo $p$, then
$g(x^{\ell})\in\langle p,h(x)\rangle$, and

$s(x^{\ell})\cdot t(x^{\ell})\in 1+\langle p,h(x)\rangle.$ (2.8)

Since $h(x)$ is a multiple factor of $f(x^{\ell})$ modulo $p$,
$f(x^{\ell})\in\langle p,h^{2}(x)\rangle.$ We have to show that
$f(x^{\ell})\not\in\langle p^{2},h(x)\rangle$. Since $g(x^{\ell})\in\langle
p,h(x)\rangle$, we can see that

$g^{2}(x^{\ell})\in\langle p,h(x)\rangle^{2}\subset\langle p^{2},h(x)\rangle$

and

$p\cdot g(x^{\ell})\in\langle p^{2},p\cdot h(x)\rangle\subset\langle
p^{2},h(x)\rangle.$

Using Equation (2.7), we get that $f(x^{\ell})\in\langle p^{2},h(x)\rangle$ if
and only if $p\cdot s(x^{\ell})\in\langle p^{2},h(x)\rangle$. We can write
$s(x^{\ell})=h(x)\cdot u(x)+v(x),$ where $\deg v(x)<\deg h(x)$ and
$u(x),v(x)\in\mathbb{Z}[x]$. Thus, $p\cdot s(x^{\ell})\in\langle
p^{2},h(x)\rangle$ if and only if $p\cdot v(x)\in\langle p^{2},h(x)\rangle$.
As $\deg v(x)<\deg h(x)$, $p\cdot v(x)\in\langle p^{2},h(x)\rangle$ if and
only if $v(x)\in\langle p\rangle$. If $v(x)$ is a multiple of $p$, then
$s(x^{\ell})\in\langle p,h(x)\rangle$, which is a contradiction to Equation
(2.8). Therefore, $f(x^{\ell})\not\in\langle p^{2},h(x)\rangle$. Hence,
Theorem 2.B together with Lemma 2.1 implies that $p$ does not divide the index
of $f(x^{\ell})$. ∎

######  Remark 2.11.

The main idea behind the last part of the proof is that if $g(x)$ is
irreducible modulo $p$ and $\deg s(x)<\deg g(x)$, then
$\overline{g}(x^{\ell})$ and $\overline{s}(x^{\ell})$ are coprime. But
$f(x^{\ell})$ belongs to $\langle p^{2},h(x)\rangle$ if and only if
$\overline{h}(x)$ is a common divisor of $\overline{g}(x^{\ell})$ and
$\overline{s}(x^{\ell})$.

The following lemma proved in [16, Lemma 2.17] will be used in the proof of
the Theorem 1.8.

######  Lemma 2.12.

Let $\alpha$ be an algebraic integer, and let $L=\mathbb{Q}(\alpha)$ be the
field generated by it. If the minimal polynomial $f(x)$ of $\alpha$ over
$\mathbb{Q}$ is an Eisenstein polynomial with respect to the prime $p$, i.e.,
it has the form $x^{n}+a_{n-1}x^{n-1}+\cdots+a_{0},$ with
$a_{0},\cdots,a_{n-1}$ divisible by $p$ and $p^{2}\nmid a_{0},$ then the index
of $f(x)$ is not divisible by $p.$

##  3\. Proof of the main results

###### Proof of Theorem 1.1.

First, assume that $f(x^{k})$ is monogenic. Then by Lemma 2.6, $f(x)$ and
$f(x^{p})$ are also monogenic for all $p\mid k$, so (1) and (2) hold.
Furthermore, there is no prime $p$ that divides the index of $f(x^{k})$, so by
Proposition 2.3, $p^{2}\nmid f(0)$ for any prime $p$, i.e., $f(0)$ is
squarefree.

Conversely, assume that (1), (2) and (3) hold. By Proposition 2.10, if $p\nmid
k$, then (1) and (3) together imply that $p$ does not divide the index of
$f(x^{k})$. Let $p\mid k$ and write $k=p^{u}\ell$, where $p\nmid\ell$. Using
Corollary 2.5 and (2), we see that $p$ does not divide the index of
$f(x^{p^{u}})$. Now, using (3) and Proposition 2.10, we get that $p$ does not
divide the index of $f(x^{k})$. Thus, $f(x^{k})$ is monogenic. ∎

###### Proof of Corollary 1.3.

Let $p$ be a prime divisor of the index of $f(x^{k})$. Suppose, that $p\nmid
f(0)$. If $p\nmid k$, then by Proposition 2.10, $p$ divides the index of
$f(x)$, which contradicts (1). On the other hand, if $p\mid k$, then we can
write $k=p^{u}\ell$, where $p\nmid\ell$. By Proposition 2.10, $p$ divides the
index of $f(x^{p^{u}})$, hence, by Corollary 2.5, $p$ divides the index of
$f(x^{p})$. This contradicts (2), so $p$ must be a divisor of $f(0)$. ∎

###### Proof of Corollary 1.4.

One can easily see that the monogenity of an irreducible polynomial of the
type $f(x)=x^{d}+A\cdot h(x)\in\mathbb{Z}[x],$ with $d>1,$ and $\deg h(x)<d$,
implies that $A$ is squarefree. Let $p\mid A$. Then $f(x)\in\langle
p,x^{2}\rangle$, so by Corollary 2.2 and Theorem 2.B, the monogenity of $f(x)$
implies that $A$ is squarefree. In this situation, $f(x)$ is an Eisenstein
polynomial with respect to $p$ and hence by Lemma 2.12, $p$ does not divide
the index of $f(x)$. Therefore the proof is complete keeping Theorem 1.1 in
mind. ∎

###### Proof of Theorem 1.5.

Set $\deg f(x)=d.$ Using Lemma 2.8, we get

$D(f(x^{p^{u}}))=\pm D(f(x))^{p^{u}}\cdot p^{udp^{u}}\cdot f(0)^{p^{u}-1}.$

(1) In view of Lemma 2.9, Equation (1.1) and the hypothesis that $f(x)$ is
monogenic, we conclude that any prime divisor of the index of $f(x^{p^{u}})$
is either $p$ or a divisor of $f(0)$. If $|f(0)|=1$, then the index of
$f(x^{p^{u}})$ is equal to $p^{s}$, for some non-negative integer $s$. Let
$q\neq p$ be a prime divisor of $f(0)$. Since $f(x)$ is monogenic and $f(0)$
is squarefree, Proposition 2.10 implies that $q$ does not divide the index of
$f(x^{p^{u}})$. Hence, $p$ is the only possible prime divisor of the index of
$f(x^{p^{u}})$.
(2) Using Corollary 2.5, it is enough to prove that $p$ does not divide the
index of $f(x^{p})$ if and only if $\frac{f(x^{p})-f(x)^{p}}{p}$ is coprime to
$f(x)$ modulo $p$. Let
$\overline{f}(x)=\displaystyle\prod_{i=1}^{t}\overline{g}^{e_{i}}_{i}(x)$ be
the factorization of $\overline{f}(x)$ into a product of powers of distinct
irreducible polynomials $\overline{g}_{i}(x)$, where each
$g_{i}(x)\in\mathbb{Z}[x]$ monic. Here, $f(x^{p})\in\langle
p,g^{2}_{i}(x)\rangle$ for all $1\leq i\leq t$. We can write

$f(x)=\displaystyle\prod_{i=1}^{t}g_{i}^{e_{i}}(x)+p\cdot a(x),$

for some $a(x)\in\mathbb{Z}[x]$. Then

$f(x)^{p}=\bigg{(}\displaystyle\prod_{i=1}^{t}g_{i}^{e_{i}}(x)+p\cdot
a(x)\bigg{)}^{p}=\bigg{(}\displaystyle\prod_{i=1}^{t}g_{i}^{e_{i}}(x)\bigg{)}^{p}+p^{2}\cdot
t(x),$

for some $t(x)\in\mathbb{Z}[x]$. Therefore, $f(x)^{p}\in\langle
p^{2},g_{i}(x)\rangle$ for each $1\leq i\leq t.$ Notice, that
$f(x^{p})\in\langle p^{2},g_{i}(x)\rangle$ if and only if
$f(x^{p})-f(x)^{p}\in\langle p^{2},g_{i}(x)\rangle.$ Set
$f(x^{p})-f(x)^{p}=p\cdot c(x),$ for some $c(x)\in\mathbb{Z}[x]$. Then,
$f(x^{p})\in\langle p^{2},g_{i}(x)\rangle$ if and only if $p\cdot
c(x)\in\langle p^{2},g_{i}(x)\rangle$. Since $g_{i}(x)\in\mathbb{Z}[x]$ is
monic and irreducible modulo $p$ for each $i$, it is clear that $p\cdot
c(x)\in\langle p^{2},g_{i}(x)\rangle$ if and only if $c(x)=g_{i}(x)\cdot
u(x)+p\cdot v(x)$, for certain $u(x),v(x)\in\mathbb{Z}[x]$. Equivalently,
$f(x^{p})\in\langle p^{2},g_{i}(x)\rangle$ if and only if
$\frac{f(x^{p})-f(x)^{p}}{p}$ is divisible by $g_{i}(x)$ modulo $p$. Thus, we
get that $f(x^{p})\not\in\langle p^{2},g_{i}(x)\rangle$ for any $1\leq i\leq
t$ if and only if $\frac{f(x^{p})-f(x)^{p}}{p}$ is coprime to $f(x)$ modulo
$p$. Using Corollary 2.2, $f(x^{p})\not\in\langle p,g_{i}(x)\rangle^{2}$ for
any $1\leq i\leq t$ if and only if $\frac{f(x^{p})-f(x)^{p}}{p}$ is coprime to
$f(x)$ modulo $p$. Hence, our result follows from Theorem 2.B. ∎

###### Proof of Theorem 1.8.

Let us start by proving that any prime divisor $p$ of $k$ does not divide the
index of the irreducible polynomial $f(x^{p})$ if and only if (2) holds. Let
$p$ be a prime divisor of $k$, and write

$\displaystyle f(x^{p})=$ $\displaystyle\big{(}x^{d}+A\cdot h(x)-A\cdot
h(x)\big{)}^{p}+A\cdot h(x^{p})$ $\displaystyle=$
$\displaystyle\big{(}f(x)-A\cdot h(x)\big{)}^{p}+A\cdot h(x^{p})$
$\displaystyle=$ $\displaystyle f(x)^{p}+(-1)^{p}A^{p}\cdot h(x)^{p}+p\cdot
f(x)\cdot r(x)+A\cdot h(x^{p}),$

where

$r(x)=\frac{1}{p}\bigg{(}\displaystyle\sum_{i=1}^{p-1}{p\choose
i}f^{i-1}(x)\big{(}-A\cdot h(x)\big{)}^{p-i}\bigg{)}.$

This implies, that

$\frac{f(x^{p})-f(x)^{p}}{p}=\frac{\big{(}-A\cdot h(x)\big{)}^{p}+p\cdot
f(x)\cdot r(x)+A\cdot h(x^{p})}{p}.$

Therefore, $\frac{f(x^{p})-f(x)^{p}}{p}$ is coprime to $f(x)$ modulo $p$ if
and only if $\frac{1}{p}[(-A\cdot h(x))^{p}+A\cdot h(x^{p})]$ is coprime to
$f(x)$ modulo $p$. Thus, using Theorem 1.5, $p$ does not divide the index of
$f(x^{p})$ if and only if $\frac{1}{p}[(-A\cdot h(x))^{p}+A\cdot h(x^{p})]$ is
coprime to $f(x)$ modulo $p.$

Using Lemma 2.6, the monogenity of $f(x^{k})$ implies that both (1) and (2)
hold. Conversely, suppose that both $(1)$ and $(2)$ hold. If $|A|=1$, then the
result follows from Corollary 1.3. Let $p$ be a prime dividing $A$. If
$p^{2}\mid A$ then Proposition 2.3 shows that $p$ divides the index of
$f(x^{p})$, a contradiction to (2). Thus, $A$ must be squarefree. Since
$|h(0)|=1$ and $A$ is squarefree, we can see that $f(x^{k})$ is an Eisenstein
polynomial with respect to $p$. Therefore, by Lemma 2.12, $p$ does not divide
the index of $f(x^{k})$. This shows, that the index of $f(x^{k})$ and $A$ are
coprime. Hence, Corollary 1.3 implies that $f(x^{k})$ is monogenic. ∎

##  4\. Infinite examples

In this section, we introduce some well-known infinite parametric families of
polynomials for which Theorem 1.1 can be applied efficiently. The first
example is the family of pure polynomials obtained by choosing $f(x)=x-A$,
where $A$ is an integer. In this case, our theorems imply the following well-
known result (see [5, Theorem 1.3]).

######  Proposition 4.1.

Let $k>1$ be an integer and $f(x)=x-A\in\mathbb{Z}[x]$. If $f(x^{k})$ is
irreducible, then $f(x^{k})$ is monogenic if and only if

  1. (i)

$p^{2}\nmid A^{p}-A$ for any prime $p\mid k$ and

  2. (ii)

$A$ is squarefree.

###### Proof.

The polynomial $f(x)=x-A$ is always monogenic, so (1) of Theorem 1.1 is true
for any $A\in\mathbb{Z}$. In view of Theorem 1.1, it is enough to prove that
(2) of Theorem 1.1 and (i) are equivalent. Here $f(x^{p})\equiv(x-A)^{p}\mod
p$. Theorem 2.B together with Corollary 2.2 imply that $p$ divides the index
of $f(x^{p})$ if and only if the remainder of $f(x^{p})$ divided by $x-A$,
which is $A^{p}-A$, is a multiple of $p^{2}$. This completes the proof. ∎

The next example is the family of simplest cubic polynomials of the form

$f(x)=x^{3}-mx^{2}-(m+3)x-1.$

In some special cases, we can deal with this family by using Theorem 1.1.

######  Proposition 4.2.

Let $m\in\mathbb{Z}$ and $f(x)=x^{3}-mx^{2}-(m+3)x-1$. Let $k$ be a positive
integer such that $f(x)$ is reducible modulo every odd prime $p\mid k$. Then
$f(x^{k})$ is monogenic if and only if

  1. (i)

$f(x)$ is monogenic, and

  2. (ii)

$m\not\equiv\dfrac{r^{3p}-3r^{p}-1}{r^{p}(r^{p}+1)}\mod p^{2}$, for any odd
prime $p\mid k$ and $r=1,\ldots,p-2$.

###### Proof.

We note, that $f(x^{k})$ is always irreducible in $\mathbb{Z}[x]$ (see [13,
Lemma 3.1]). Referring back to Theorem 1.1, we only have to prove that (2) of
Theorem 1.1 is equivalent to (ii). Let $p$ be an odd prime. The Galois group
of $f(x)$ is the cyclic group of order three, so the reducibility of $f(x)$
modulo $p$ implies that it splits into three linear factors. Therefore, the
multiple irreducible factors of $f(x^{p})$ modulo $p$ are of the form $x-r$.
Since $f(0)=-1$ and $f(-1)=1$, then $r\not\equiv 0,-1\mod p$. By Theorem 2.B
and Corollary 2.2, $p$ divides the index of $f(x^{p})$ if and only if
$f(x^{p})\in\langle p^{2},x-r\rangle$ for some $r=1,\ldots,p-2$, i.e., if and
only if $f(r^{p})\equiv 0\mod{p^{2}}$. Since $r\not\equiv 0,-1\mod{p}$, we
have $f(r^{p})=r^{3p}-mr^{2p}-(m+3)r^{p}-1\equiv 0\mod{p^{2}}$ if and only if

$m\equiv\dfrac{r^{3p}-3r^{p}-1}{r^{p}(r^{p}+1)}\mod{p^{2}}.$

Finally, it can be easily checked that for $p=2$, $f(x)$ is always irreducible
modulo 2, and $f(x^{2})$ is never contained in $\langle 2,f(x)\rangle^{2}$, so
$2$ can never divide the index of $f(x^{k})$. ∎

######  Remark 4.3.

In Proposition 4.2, the condition $f(x)$ is reducible modulo every odd prime
$p\mid k$ can not be dropped. For example, if $k=13$ and $m\equiv 6\mod{13}$,
then $f(x)$ is irreducible modulo $13$, however, if $m=71$, then both (i) and
(ii) are true, but 13 divides the index of $f(x^{k})$. For the sake
completeness, see [13, Lemma 3.3], in which the author gives a
characterization of the monogenity of $f(x^{p})$, where
$f(x)=x^{3}-mx^{2}-(m+3)x-1$ is irreducible modulo $p$.

In the last example, we apply Theorem 1.8 to $f(x)=x^{d}+A(Bx+1)^{m}$. The
monogenity of $f(x)$ is investigated in [15].

######  Proposition 4.4.

Let $A$ and $B$ be integers and $d$ and $m$ positive integers such that $d>m$.
Assume, that $\gcd(d,mB)=1.$ Let $f(x)=x^{d}+A(Bx+1)^{m}$ and let $k$ be a
positive integer such that $f(x^{k})$ is irreducible. Then $f(x^{k})$ is
monogenic if and only if the following statements hold:

  1. (1)

Both $A$ and $d^{d}+(-1)^{d+m}B^{d}m^{m}(d-m)^{d-m}A$ are squarefree.

  2. (2)

For any prime $p$ dividing $k$,

$\frac{\big{(}-A(Bx+1)^{m}\big{)}^{p}+A(Bx^{p}+1)^{m}}{p}$

is coprime to $f(x)$ modulo $p.$

Furthermore, if $\operatorname{rad}(k)\mid\operatorname{rad}(A)$, then
$f(x^{k})$ is monogenic if and only if both $A$ and
$d^{d}+(-1)^{d+m}B^{d}m^{m}(d-m)^{d-m}A$ are squarefree.

###### Proof.

If $A\neq\pm 1$ is a squarefree integer, then $f(x^{k})$ is an Eisenstein
polynomial with respect to a prime divisor of $k$, thus $f(x^{k})$ is
irreducible. This implies, that there exist an infinite number of irreducible
polynomials of the form $f(x^{k})=x^{dk}+A(Bx^{k}+1)^{m}$. By [15, Theorem
1.1], $f(x)$ is monogenic if and only if both $A$ and
$d^{d}+(-1)^{d+m}C^{d}m^{m}(d-m)^{d-m}A$ are squarefree. Set
$h(x)=(Bx+1)^{m}$, then $h(0)=1$ and $\deg h(x)<d$, and our result follows
from Theorem 1.8. ∎

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