# A parametrization of nonassociative cyclic algebras of prime degree

Monica Nevins  Department of Mathematics and Statistics, University of Ottawa,
Ottawa, Canada K1N 6N5<EMAIL_ADDRESS>and  Susanne Pumplün  School of
Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD
<EMAIL_ADDRESS>
###### Abstract.

We determine and explicitly parametrize the isomorphism classes of
nonassociative quaternion algebras over a field of characteristic different
from two, as well as the isomorphism classes of nonassociative cyclic algebras
of odd prime degree when the base field contains a primitive $m$th root of
unity. In the course of doing so, we prove that any two such algebras can be
isomorphic only if the cyclic field extension and the chosen generator of the
Galois group are the same. As an application, we give a parametrization of
nonassociative cyclic algebras of prime degree over a local nonarchimedean
field $F$, which is entirely explicit under mild hypotheses on the residual
characteristic. In particular, this gives a rich understanding of the
important class of nonassociative quaternion algebras up to isomorphism over
nonarchimedean local fields.

###### Key words and phrases:

Nonassociative division algebras, nonassociative cyclic algebras.

###### 2020 Mathematics Subject Classification:

17A35, 17A99

The first author’s research is supported NSERC Discovery Grant
RGPIN-2020-05020.

##  1\. Introduction

The general study of the classification and construction of nonassociative
algebras is a classical subject, dating back to pioneering works of [AlbI1942,
AlbII1942, Di1935]. It is a rich area, including such important examples as
Jordan algebras which go back to [Pas1933], and semifields (_i.e._ ,
nonassociative division algebras over finite fields).

In this paper, we give a parametrization of the isomorphism classes of a
particular class of nonassociative algebras first defined (slightly more
generally) as Sandler semifields in [San62], and later studied over any base
field (_e.g._ in [St2014]): nonassociative cyclic algebras of degree $m$
(Definition 2.1). They are defined by a cyclic Galois field extension $K/F$ of
degree $m$, a generator $\sigma$ of $\mathrm{Gal}(K/F)$, and an element $a\in
K\smallsetminus F$, and are denoted $(K/F,\sigma,a)$. In particular, when the
degree $m$ is prime, and $F$ contains a primitive $m$th root of unity, a
nonassociative cyclic algebra of degree $m$ over $F$ is necessarily a division
algebra [St2014]. These algebras are a direct generalization of associative
cyclic algebras which, over local and global fields $F$, generate the Brauer
group of (associative) central simple algebras over $F$. Associative central
simple algebras similarly arise as the key objects in the theorems of
Merkurjev and Suslin on the $m$-torsion of Brauer groups over arbitrary
fields.

After deriving our explicit parametrizations (in the quadratic case in Theorem
4.2 and in the case of extensions of prime degree $m$, assuming $F$ contains a
primitive $m$th root of unity in Theorem 5.5), we go on to apply our
parametrization to nonassociative cyclic algebras over local nonarchimedean
fields $F$. These are the finite algebraic extensions of the $p$-adic numbers
$\mathbb{Q}_{p}$, together with Laurent series over a finite field. Local
nonarchimedean fields lie at the intersection of real and finite fields: they
are complete locally compact fields that are suitable for use in analysis, but
retain abundant number-theoretic properties related to their finite residue
fields. We exploit these features to give a classification that is elegant and
completely explicit under mild hypotheses on the field: for nonassociative
quaternion algebras, the relevant results are Theorems 4.4, 4.5 and 4.6; for
higher prime degree, our general result follows from Proposition 5.6 and we
present in detail the case of $m=3$ in Theorem 5.7.

In the course of deriving our results, we also prove the following
counterintuitive result (Theorem 5.1), valid for nonassociative cyclic
algebras of arbitrary degree (and in direct contrast with the associative
case).

######  Theorem.

Let $F$ be an arbitrary field and let $m\geq 3$ be the degree of a cyclic
Galois extension $K/F$. For any two distinct generators
$\sigma_{1}\neq\sigma_{2}$ of the Galois group $\mathrm{Gal}(K/F)$, and for
any $a_{1},a_{2}\in K\smallsetminus F$, we have

$(K/F,\sigma_{1},a_{1})\not\cong(K/F,\sigma_{2},a_{2}).$

Given the importance of associative cyclic algebras in the Brauer group of
$F$, it is natural to expect that nonassociative cyclic algebras (which are
pivotal examples of semiassociative algebras) will play a similarly central
role in the recently defined semiassociative Brauer monoid [BHMRV2024,
Pum2024]. Nonassociative cyclic algebras also recently appeared as first
examples of residue $m$-hyperrings [Row2024, Example 4.10 (i), Remark 4.11].

Furthermore, in the special case that $m$ is equal to 2, the nonassociative
cyclic algebras over $F$ are exactly all the four-dimensional unital algebras
over $F$ which have a separable quadratic field extension of $F$ contained in
their nucleus [Wat1987]. They are also identical to the nonassociative
quaternion (division) algebras that are obtained by a generalized Cayley-
Dickson doubling [AsPu2006].

In general, the classification of $n$-dimensional nonassociative algebras is a
wild problem. Rough classifications up to the derivation types of the
algebras, or up to their automorphism groups, are more tractable, and are
possible for small dimensions, fixed base fields, or certain families of
nonassociative algebras. Classifications up to isotopy, or even up to
isomorphism, are often difficult even for small dimensions and special base
fields. Investigations of special classes of small dimensional algebras
usually involve structure constants. For a comprehensive recent literature
review, see [Ka2024, Section 2.1].

Three- and four-dimensional nonassociative division algebras over a $p$-adic
field were roughly classified up to isotopy in [DeaGra2011] and Limburg
[Lim2006] by studying the determinant of the matrix of their left
multiplication and classifying it up to isometry. Unfortunately, this
classification does not reveal the algebraic structure for some of the classes
of algebras listed.

To our knowledge, ours are some of the first known parametrizations of a large
class of nonassociative division algebras over nonarchimedean local fields
$F$, and our work invites several interesting next directions.

For one, there should be a relationship between the nonassociative cyclic
algebras arising from unramified extensions and those arising from the
corresponding extensions of the finite residue field, through localization.
Note that nonassociative cyclic algebras are special cases of Petit algebras
[Pet1966, Pet1968, LawSh2013], and over finite base fields isotopic to Jha-
Johnson semifields [JJ989], which form one of the largest known class of
semifields. However, it is evident from examples that this localization
depends on the parameters in ways that are not evident from the high-level
theory.

We also believe that the explicit nature of our parameterization will help in
tackling concrete open problems in the new theory of semiassociative algebras.
We note that nonassociative cyclic algebras are also employed in the
construction of space-time codes which are used in digital data transmission,
and in linear code constructions; see, for example, [Pum2016, PumSt2015,
PuSt2015]. It would be very interesting to explore how to exploit the
essentially binary nature of nonassociative cyclic algebras over
$\mathbb{F}_{2}(\\!(t)\\!)$, for example, to the development of more efficient
linear codes.

The paper is organized as follows. We set our notation and collect some main
results on nonassociative cyclic algebras and nonarchimedean local fields in
Sections 2 and 3. In Section 4 we prove our parametrization (a classification
up to isomorphism) of nonassociative quaternion algebras, first over general
fields of characteristic different from $2$ (Theorem 4.2), and then
specialized to nonarchimedean local fields, including the characteristic $2$
case, which has a very different flavour (Theorem 4.6). In particular, we have
thus achieved full parametrizations of all exactly all four-dimensional unital
algebras over $F$ which have a separable quadratic field extension of $F$
contained in their nucleus.

Section 5 contains the two main results of this paper (Theorems 5.1 and 5.5).
We then derive the parametrization over local nonarchimedean fields of
residual characteristic different from $m$ (and containing $\mu_{m}$)
employing Lemma 5.6 and in particular for degree three in Section 5.4. We
conclude in Section 6 with a short exploration of the case of degree $4$
nonassociative cyclic algebras over a local nonarchimedean field such that
$p\neq 2$.

##  2\. Notation and background

We use $R^{\times}$ to denote the group of invertible elements of a unital
associative ring $R$, and denote by $(R^{\times})^{n}$ the subgroup
$\\{x^{n}\mid x\in R^{\times}\\}$. Let $F$ be a field and let $\mu_{m}$ be the
group of $m$th roots of unity in a fixed algebraic closure of $F$, when the
characteristic of $F$ does not divide $m$.

###  2.1. Nonassociative algebras

An $F$-vector space $A$ is an _algebra_ over $F$ if there exists an
$F$-bilinear map $A\times A\to A$, $(x,y)\mapsto x\cdot y$. We denote this
_multiplication_ in $A$ simply by the juxtaposition $xy$. An algebra $A$ is
called _unital_ if there is an element in $A$, denoted by 1, such that
$1x=x1=x$ for all $x\in A$. We will only consider unital finite-dimensional
nonzero algebras.

Define the _associator_ of $A$ by $[x,y,z]=(xy)z-x(yz)$. The left, middle and
right nucleus of $A$ are defined as ${\rm Nuc}_{l}(A)=\\{x\in
A\,|\,[x,A,A]=0\\}$, ${\rm Nuc}_{m}(A)=\\{x\in A\,|\,[A,x,A]=0\\}$ and ${\rm
Nuc}_{r}(A)=\\{x\in A\,|\,[A,A,x]=0\\}$, respectively. These are associative
subalgebras of $A$ and their intersection ${\rm Nuc}(A)=\\{x\in
A\,|\,[x,A,A]=[A,x,A]=[A,A,x]=0\\}$ is the nucleus of $A$. The center of $A$
is ${\rm C}(A)=\\{x\in A\,|\,x\in\text{Nuc}(A)\text{ and }xy=yx\text{ for all
}y\in A\\}$.

An algebra $A\not=0$ is called a _division algebra_ if for any $a\in A$,
$a\not=0$, the left multiplication with $a$, $L_{a}(x)=ax$, and the right
multiplication with $a$, $R_{a}(x)=xa$, are bijective. If $A$ has finite
dimension over $F$, $A$ is a division algebra if and only if $A$ has no zero
divisors [Sch1995, pp. 15, 16].

###  2.2. Nonassociative cyclic algebras

We now define our principal objects of study; we adapt the definition from
[St2014], where the opposite algebras were considered instead.

######  Definition 2.1.

Let $K/F$ be a cyclic Galois field extension of degree $m$ and let $\sigma$ be
a generator of $\mathrm{Gal}(K/F)$. Then for any $a\in K^{\times}$, the
_nonassociative cyclic algebra $(K/F,\sigma,a)$ of degree $m$_ is the unital
algebra of dimension $m^{2}$ defined by

$(K/F,\sigma,a)=\bigoplus_{s=0}^{m-1}Kt^{s}$

with $F$-bilinear multiplication defined for $0\leq s,s^{\prime}<m$ and
$k,k^{\prime}\in K$ by

(2.1)
$(kt^{s})(k^{\prime}t^{s^{\prime}})=\begin{cases}k\sigma^{s}(k^{\prime})t^{s+s^{\prime}}&\text{if
$s+s^{\prime}<m$;}\\\ k\sigma^{s}(k^{\prime})at^{s+s^{\prime}-m}&\text{if
$s+s^{\prime}\geq m$.}\end{cases}$

Note that for all $a\in K\smallsetminus F$ the algebra $(K/F,\sigma,a)$ is not
associative, not even power-associative, since for example
$t(t^{m})=ta=\sigma(a)t$ but $(t^{m})t=at$. If we instead choose $a\in
F^{\times}$, then the definition above yields the classical central simple
associative cyclic algebra $(K/F,\sigma,a)$ of degree $m$ [Jac1996, §1.4]. In
the following, we call a nonassociative cyclic algebra $(K/F,\sigma,a)$ that
is not associative (_i.e._ $a\in K\smallsetminus F$) sometimes also a _proper_
nonassociative cyclic algebra.

An equivalent way to define $(K/F,\sigma,a)$ involves twisted polynomials: Let
$f(t)=t^{m}-d\in R=K[t;\sigma]$. Then $R_{m}=\\{g\in K[t;\sigma]\,|\,{\rm
deg}(g)<m\\}$ together with the usual addition and the multiplication defined
via $g\circ h=gh\,\,{\rm mod}_{r}f$, where the right hand side is simply the
remainder of $gh$ after division by $f$ of the right, is the nonassociative
cyclic algebra $(K/F,\sigma,a)$, also denoted by $R/R(t^{m}-a)$ as it is a
special case of a Petit algebra [Pet1966]. Note that when ${\rm deg}(g)+{\rm
deg}(h)<m$, the multiplication in $(K/F,\sigma,a)$ is the same as the
multiplication in $R$. If $a\in F^{\times}$ then $(K/F,\sigma,a)$ is the
classical central simple associative cyclic algebra $(K/F,\sigma,a)$ which can
also be written as the quotient algebra $K[t;\delta]/(f)$ [Pet1966, (7),(9),
(10)], [Jac1996, p. 19]. This justifies the notation $R/R(t^{m}-a)$ introduced
by Petit.

We record some quick properties.

######  Lemma 2.2.

[Pet1966] Let $A=(K/F,\sigma,a)$ with $a\in K\smallsetminus F$. Then $C(A)=F$,
${\rm Nuc}_{l}(A)={\rm Nuc}_{m}(A)=K$ and ${\rm Nuc}_{r}(A)=\\{g\in
R/Rf\,|\,fg\in Rf\\}$.

Note that $\\{g\in R/Rf\,|\,fg\in Rf\\}$ can be identified as the
$0$-eigenspace of the right multiplication operator $R_{f}$, also called the
_eigenspace of $f$_ [Pet1966].

In particular when $m$ is prime we must have ${\rm
Nuc}_{r}((K/F,\sigma,a))=K$. We also know that $(K/F,\sigma,a)$ is a division
algebra if and only if ${\rm Nuc}_{r}((K/F,\sigma,a))$ is a division algebra,
_e.g._ see [GoTor2014, Proposition 4].

Nonassociative cyclic algebras of degree 2 are also called _nonassociative
quaternion algebras_ , and were first described by Dickson [Di1935]. They can
also be constructed by a generalized Cayley-Dickson process ${\rm Cay}(K,a)$
[AsPu2006, Lemma 1], as it coincides with the specialization to $m=2$ of
Definition 2.1.

###  2.3. Properties and isomorphisms

Let us now summarize some needed results from the literature.

######  Theorem 2.3.

[Pet1966] Suppose either that $m=2$ or $3$, or that $m\geq 5$ is a prime such
that $F$ contains a primitive $m$th root of unity. Let $a\in K\smallsetminus
F$. Then $(K/F,\sigma,a)$ is a division algebra.

The case $m=2$ is distinguished: nonassociative quaternion algebras are, up to
isomorphism, the only four-dimensional unital division algebras over a field
$F$ that have a quadratic field extension $K/F$ in their nucleus, and which
are not associative [Wat1987, Theorem 1]. We thus put a special emphasis on
them in this paper.

When $m$ is not prime, we have the following statement.

######  Theorem 2.4.

[St2014, Theorem 4.4, Corollary 4.5] Suppose that $(K/F,\sigma,a)$ is a
nonassociative cyclic algebra where $K/F$ is of degree $m$. If $a$ does not
lie in a proper subfield of $K/F$, then $(K/F,\sigma,a)$ is a division
algebra.

###### Proof.

If $1,a,\cdots,a^{m-1}$ are linearly independent over $F$, then
$(K/F,\sigma,a)$ is a division algebra [St2014, Theorem 4.4]. The elements
$1,a,\cdots,a^{m-1}$ are linearly independent over $F$ if and only if the
minimal polynomial of $a$ over $F$ has degree at least $m$, whence $F(a)/F$
has degree at least $m$. Thus this is condition is equivalent to having
$K=F(a)$ and thus that $a$ lies in no proper subfield of $K$. ∎

We now turn to the question of when two such algebras are isomorphic.

######  Lemma 2.5.

Suppose $K,K^{\prime}$ are two distinct cyclic Galois extensions of a field
$F$. Then for any generators $\sigma,\sigma^{\prime}$ of their respective
Galois groups, and any $a\in K\smallsetminus F$, $a^{\prime}\in
K^{\prime}\smallsetminus F$, we have

$(K/F,\sigma,a)\not\cong(K^{\prime}/F,\sigma^{\prime},a^{\prime}).$

The proof is immediate, since isomorphic algebras will have the same left
nuclei. In the rest of the paper, we will rely on the following strong
classification theorem [BrPum2018, Corollary 32].

######  Theorem 2.6.

Let $K$ be a cyclic Galois extension of $F$ and let $\sigma$ be a generator of
$\mathrm{Gal}(K/F)$. For $a,b\in K\smallsetminus F$ we have that
$(K/F,\sigma,a)\cong(K/F,\sigma,b)$ if and only if there exists some
$\tau\in\mathrm{Gal}(K/F)$ such that

(2.2) $a\in\tau(b)N_{K/F}(K^{\times}).$

For each such extension $K$, we denote by $\sim$ the equivalence relation on
$K\smallsetminus F$ induced by (2.2), that is, $a\sim b$ if and only if there
is some power $i\in\mathbb{Z}$ such that
$a\in\sigma^{i}(b)N_{K/F}(K^{\times})$.

##  3\. Background on p-adic fields

A local nonarchimedean field is a locally compact, nondiscrete field equipped
with an ultrametric norm $|\cdot|$. For each prime number $p$, there are
precisely two kinds: the finite algebraic extensions of the $p$-adic numbers
$\mathbb{Q}_{p}$, which are the completion of a number field with respect to
the $p$-adic norm; and the fields $\mathbb{F}_{q}(\\!(t)\\!)$ of Laurent
series over a finite field of order $q=p^{n}$ for some $n\geq 1$, with
$|t|<1$. The former are called $p$-adic fields, and have characteristic zero.
An excellent resource for the summary given here is [Lan94, Chapter II]; for
examples, see [LMF24].

Let $F$ be a local nonarchimedean field. Its integer ring is its maximal
compact open subring $\mathcal{R}=\\{x\in F:|x|\leq 1\\}$. This ring is local,
with unique maximal ideal $\mathcal{P}=\\{x\in F:|x|<1\\}$. Its unit subgroup
of invertible elements is then
$\mathcal{R}^{\times}=\mathcal{R}\smallsetminus\mathcal{P}$. The quotient
$\kappa:=\mathcal{R}/\mathcal{P}$, called the _residue field_ of $F$, is a
finite field of order $q=p^{n}$ for some $n\in\mathbb{N}$ and prime $p$
(called the _residual characteristic_ of $F$).

######  Example 3.1.

If $F=\mathbb{F}_{q}(\\!(t)\\!)$ then $\mathcal{R}=\mathbb{F}_{q}[\\![t]\\!]$,
$\mathcal{P}=(t)$, and $\kappa=\mathbb{F}_{q}$. Similarly, if
$F=\mathbb{Q}_{p}$ then we have

$\mathbb{Q}_{p}^{\times}=\\{a=\sum_{i=N}^{\infty}a_{i}p^{i}:N\in\mathbb{N},a_{i}\in\\{0,1,\ldots,p-1\\}\\}$

with addition and multiplication computed “with carrying” as one would for
finite sums. Here the least $i$ such that $a_{i}\neq 0$ is the $p$-adic
valuation of $a$, and in this case $|a|=p^{-i}$. The $p$-adic integers
$\mathcal{R}=\mathbb{Z}_{p}$ is the set of all elements of valuation at least
$0$.

More generally, we define the valuation of $a\in F^{\times}$ by
$v(a)=\min\\{n\in\mathbb{Z}:a\in\mathcal{P}^{n}\\}$ and scale the norm so that
$|a|=q^{-v(a)}$. We fix a generator $\varpi$ of $\mathcal{P}$ for once and for
all; it satisfies $v(\varpi)=1$.

The fundamental relationship between $F$ and $\kappa$ is captured by Hensel’s
Lemma. The following general version is adapted from [Eis95, Thm 7.3].

######  Lemma 3.2 (Hensel’s Lemma).

Let $f(x)\in\mathcal{R}[x]$ be a polynomial and $f^{\prime}(x)$ its formal
derivative. If $a\in\mathcal{R}$ satisfies

$f(a)\equiv 0\mod f^{\prime}(a)^{2}\mathcal{P},$

then there exists a unique $b\in\mathcal{R}$ satisfying

$f(b)=0\quad\text{and}\quad b\equiv a\mod f^{\prime}(a)\mathcal{P}.$

######  Example 3.3.

The elements of $\kappa^{\times}$ are the roots of $f(x)=x^{q-1}-1$. For each
$\overline{a}\in\kappa^{\times}$, choose $a\in\mathcal{R}$ to be some
preimage. Since $f(a)\equiv 0\mod\mathcal{P}$ and
$f^{\prime}(a)=(q-1)a^{q-2}\in\mathcal{R}^{\times}$, Hensel’s Lemma implies
that there exists a unique $b\in\mathcal{R}$ satisfying $b^{q-1}=1$ and such
that $b\in a+\mathcal{P}$. The set of all such $b$ (sometimes called
Teichmüller lifts of elements of $\kappa^{\times}$) are distinct and form the
group of $(q-1)$th roots of unity $\mu_{q-1}$ in $F$. Note that we similarly
recover $\mu_{n}\cap F$ for any $n$ not divisible by $p$, and this group will
have order $n$ exactly when $n|(q-1)$.

It is sometimes convenient to decompose $F^{\times}$ as the direct product of
groups

(3.1) $F^{\times}\cong\mu_{q-1}\times(1+\mathcal{P})\times\mathbb{Z}$

via the map that associates to a triple $(b,u,n)$ the element $bu\varpi^{n}$.

Finite algebraic extensions $F$ are again local nonarchimedean fields and they
come in two forms. There is a unique (Galois) _unramified extension_ of each
degree $n$, which is the splitting field of $x^{q^{n}}-x$. We often denote it
$L_{n}$. Its residue field is $\kappa_{L_{n}}\cong\mathbb{F}_{q^{n}}$ and its
maximal ideal $\mathcal{P}_{L_{n}}$ is generated over $\mathcal{R}_{L_{n}}$ by
$\varpi$.

At the other extreme are the totally ramified extensions $K$ of $F$ of degree
$n$, of which there are usually several (in fact, infinitely many if the
characteristic of $F$ is $p=n$), and not all of which are Galois. They each
have the property that their residue field satisfies $\kappa_{K}\cong\kappa$
and $\varpi$ generates $\mathcal{P}_{K}^{n}$.

A general finite algebraic extension $K$ of $F$ can be uniquely factored as an
unramified extension $L/F$ of degree $f$ followed by a totally ramified
extension $K/L$ of degree $e$, the _ramification index_. We say $K/F$ is
_tamely ramified_ if $(e,p)=1$ and _wildly ramified_ if not. The totally and
tamely ramified extensions of degree $e$ are all obtained by adjoining to $F$
a root of $x^{n}-u\varpi$, for some choice of
$u\in\mathcal{R}^{\times}/(\mathcal{R}^{\times})^{e}$; these are Galois only
if $F$ contains a primitive $e$th root of unity.

By local class field theory [Lan94, XI §4], the Galois extensions of degree
$n$ of $F$ are in one-to-one correspondence with the subgroups of index $n$ of
$F^{\times}$, via the map that assigns $K/F$ to the subgroup
$N_{K/F}(K^{\times})$, which is the image of the norm map. In particular, this
implies that $F^{\times}/N_{K/F}(K^{\times})$ is a finite group of order $n$
equal to the degree of the extension.

For example, if $K=L_{n}$ is an unramified extension, then upon restriction to
the integer ring, the norm map
$N_{K/F}:\mathcal{R}_{K}^{\times}\to\mathcal{R}^{\times}$ is surjective, and
the group $F^{\times}/N_{K/F}(K^{\times})$ is represented by the elements

$\\{1,\varpi,\varpi^{2},\cdots,\varpi^{n-1}\\}.$

On the other hand, for a totally and tamely ramified extension,
$F^{\times}/N_{K/F}(K^{\times})\cong\mathcal{R}^{\times}/(\mathcal{R}^{\times})^{n}$
and Hensel’s Lemma identifies this group with
$\kappa^{\times}/(\kappa^{\times})^{n}$.

These features make classifying nonassociative cyclic algebras explicitly
quite tractable.

##  4\. Nonassociative quaternion algebras

In this section, we first derive some general results about nonassociative
quaternion algebras and then apply these to the case of local nonarchimedean
fields. As noted, these algebras exhaust the four-dimensional unital
nonassociative division algebras that contain a quadratic field extension in
their nucleus.

###  4.1. Isomorphism classes of nonassociative quaternion algebras in odd
characteristic

Throughout this section, let $F$ denote an arbitrary field of characteristic
different from two. We begin with an elementary lemma.

######  Lemma 4.1.

Suppose the characteristic of $F$ is odd, or zero. The distinct quadratic
field extensions of $F$ are given by $F(\sqrt{c})$ as $c$ runs over the
distinct nontrivial classes in $F^{\times}/(F^{\times})^{2}$.

###### Proof.

Let $K$ be a quadratic extension field of $F$, defined as the splitting field
of an irreducible quadratic polynomial $m(x)=x^{2}+ax+b\in F[x]$. Since $2\in
F^{\times}$, by the quadratic formula, the roots of $m(x)$ are

$\alpha_{\pm}=-\frac{a}{2}\pm\frac{1}{2}\sqrt{a^{2}-4b}.$

Since $a/2,1/2\in F$, it follows that $F(\alpha_{\pm})=F(\sqrt{a^{2}-4b})$ and
that this is a field extension if and only if
$c=a^{2}-4b\notin(F^{\times})^{2}\cup\\{0\\}$. The rest of the lemma follows.
∎

Applying Lemma 2.5, we deduce that if $A$ is a proper nonassociative
quaternion algebra over $F$, then there exists a _unique_ (nontrivial) coset
$c(F^{\times})^{2}\in F^{\times}/(F^{\times})^{2}$ and an element $a\in
F(\sqrt{c})\smallsetminus F$ such that

$A\cong(F(\sqrt{c})/F,\sigma,a),$

where $\sigma\in\mathrm{Gal}(F(\sqrt{c})/F)$ denotes the unique nontrivial
element of the Galois group of this quadratic extension.

We now refine this to an explicit classification of these nonassociative
cyclic algebras.

######  Theorem 4.2.

Suppose $K$ is a separable quadratic extension of a field $F$ of the form
$K=F(\sqrt{c})$, for some $c\in F^{\times}\smallsetminus(F^{\times})^{2}$ and
let $\sigma\in\mathrm{Gal}(K/F)$ be nontrivial. Let $\mathcal{C}_{K}$ denote a
set of coset representatives of $F^{\times}/N_{K/F}(K^{\times})$. Then the
distinct isomorphism classes of the nonassociative quaternion algebras with
left nucleus $K$ are represented by $(K/F,\sigma,a)$ where $a$ is chosen from
the set $\mathcal{S}(K)$ given by

$\mathcal{S}(K)=\begin{cases}\\{r\sqrt{c},r+s\sqrt{c}\mid
r\in\mathcal{C}_{K},s\in F^{\times}/\\{\pm 1\\}\\}&\text{if $-1\in
N_{K/F}(K^{\times})$}\\\ \\{r^{\prime}\sqrt{c}\mid
r^{\prime}\in\mathcal{C}_{K}/\\{\pm 1\\}\\}&\\\ \quad\cup\\{r+s\sqrt{c}\mid
r\in\mathcal{C}_{K},s\in F^{\times}/\\{\pm 1\\}\\}&\text{if $-1\notin
N_{K/F}(K^{\times})$.}\end{cases}$

###### Proof.

We begin by noting that if $-1\notin N_{K/F}(K^{\times})$, then the quotient
group $F^{\times}/N_{K/F}(K^{\times})$ contains a class represented by $-1$,
whence the normal subgroup we denote $\\{\pm 1\\}$; this defines the
expression $\mathcal{C}_{K}/\\{\pm 1\\}$.

Since $S\subset K\smallsetminus F$, each $a\in S$ defines a nonassociative
cyclic algebra $(K/F,\sigma,a)$ with left nucleus $K$. By Theorem 2.6, two
nonassociative cyclic algebras $(K/F,\sigma,a)$ and $(K/F,\sigma,b)$ are
isomorphic if and only if $a\sim b$, that is, if

$a\in bN_{K/F}(K^{\times})\cup\sigma(b)N_{K/F}(K^{\times}).$

What we will show is that for every $b\in K\smallsetminus F$, there exists a
unique choice of $a\in\mathcal{S}(K)$ such that either $b\in
aN_{K/F}(K^{\times})$ or $\sigma(b)\in aN_{K/F}(K^{\times})$. We begin with
existence.

Let $b\in K\smallsetminus F$. Then there exist $b_{0}\in F,b_{1}\in
F^{\times}$ such that $b=b_{0}+b_{1}\sqrt{c}$.

If $b_{0}=0$, then since $b_{1}\in F^{\times}$, we may write $b_{1}=rn$ for a
unique choice of $r\in\mathcal{C}_{K}$ and $n\in N_{K/F}(K^{\times})$, whence
$b\in r\sqrt{c}N_{K/F}(K^{\times})$.

If $b_{0}\in F^{\times}$, then there exists a unique $r\in\mathcal{C}_{K}$ and
$n\in N_{K/F}(K^{\times})$ such that $b_{0}=rn$. Setting $s=b_{1}n^{-1}$, we
have

$b=(r+s\sqrt{c})n\in(r+s\sqrt{c})N_{K/F}(K^{\times}).$

Since $\sigma(r+s\sqrt{c})=r-s\sqrt{c}$, the elements $r\pm s\sqrt{c}$ always
represent the same isomorphism class of an algebra. On the other hand, since
$\sigma(r\sqrt{c})=-r\sqrt{c}$, $\sigma$ preserves the class
$r\sqrt{c}N_{K/F}(K^{\times})$ if and only if $-1\in N_{K/F}(K^{\times})$, in
which case each of these elements represent pairwise nonisomorphic algebras.
If $-1\notin N_{K/F}(K^{\times})$, then instead only the elements
$r^{\prime}\sqrt{c}$, for $r^{\prime}\in\mathcal{C}_{K}/\\{\pm 1\\}$, are
pairwise nonisomorphic. ∎

For use in practice, we briefly offer an alternative parametrization of these
nonassociative quaternion algebras.

######  Corollary 4.3.

In the setting of Theorem 4.2, we may alternatively parametrize the distinct
isomorphism classes of nonassociative quaternion algebras with left nucleus
$K$ by elements of the set $\mathcal{S}^{\prime}(K)$, where

$\mathcal{S}^{\prime}(K)=\begin{cases}\\{t+r\sqrt{c}\mid
r\in\mathcal{C}_{K},t\in\\{0\\}\cup F^{\times}/\\{\pm 1\\}\\}&\text{if $-1\in
N_{K/F}(K^{\times})$};\\\ \\{t+r\sqrt{c},\;r^{\prime}\sqrt{c}\mid
r\in\mathcal{C}_{K},r^{\prime}\in\mathcal{C}_{K}/\\{\pm 1\\},t\in
F^{\times}/\\{\pm 1\\}\\}&\text{if $-1\notin
N_{K/F}(K^{\times})$}.\end{cases}$

###### Proof.

The elements of the form $r\sqrt{c}$ in $\mathcal{S}(K)$ and
$\mathcal{S}^{\prime}(K)$ are in bijection. Suppose
$r+s\sqrt{c}\in\mathcal{S}(K)$. Then $s=r^{\prime}n$ for a unique
$r^{\prime}\in\mathcal{C}_{K}$ and $n\in N_{K/F}(K^{\times})$. Therefore with
$t=rn^{-1}$ we have

$r+s\sqrt{c}=(rn^{-1}+r^{\prime}\sqrt{c})n\in(t+r^{\prime}\sqrt{c})N_{K/F}(K^{\times}).$

Since $s$ was only defined as an element of $F^{\times}/\\{\pm 1\\}$, the same
is true of $t$, whence the bijection. ∎

###  4.2. Case of local nonarchimedean fields of odd residual characteristic

In this section, let $F$ be a nonarchimedean local field of residual
characteristic $p\neq 2$; that is, we exclude the finite algebraic extensions
of $\mathbb{Q}_{2}$ and of $\mathbb{F}_{2}(\\!(t)\\!)$. We provide a fully
explicit parametrization of the nonassociative quaternion algebras in this
case, based on Theorem 4.2.

First note that the square classes in $F^{\times}$ are represented by

$F^{\times}/(F^{\times})^{2}=\\{1,\varepsilon,\varpi,\varepsilon\varpi\\}$

where
$\varepsilon\in\mathcal{R}^{\times}\smallsetminus(\mathcal{R}^{\times})^{2}$
is a fixed nonsquare element of valuation zero, which may be thought of as a
lift to $\mathcal{R}^{\times}$ of a nonsquare element of the residue field via
Hensel’s Lemma 3.2. Therefore there are exactly three distinct quadratic
extensions of $F$ in this case:

  * •

$L_{2}=F(\sqrt{\varepsilon})$, the unique unramified quadratic extension;

  * •

$K_{\varpi}=F(\sqrt{\varpi})$, a ramified extension;

  * •

$K_{\varepsilon\varpi}=F(\sqrt{\varepsilon\varpi})$, a second ramified
extension.

Secondly, note that local class field theory gives
$F^{\times}/N_{K/F}(K^{\times})\cong\mathrm{Gal}(K/F)$. Explicitly, when $K/F$
is quadratic, this means we may take our set of representatives to be

$\mathcal{C}_{K}=\begin{cases}\\{1,\varpi\\}&\text{if $K=L_{2}$ is unramified
over $F$};\\\ \\{1,\varepsilon\\}&\text{if $K/F$ is ramified.}\end{cases}$

Thirdly: when $K=L_{2}$ is unramified, $-1\in\mathcal{R}^{\times}\subset
N_{K/F}(K^{\times})$, independent of $p>2$. When $K/F$ is ramified, however,
then $-1\in N_{K/F}(K^{\times})$ if and only if $-1\in(F^{\times})^{2}$. Since
$p\neq 2$, Hensel’s Lemma 3.2 implies this occurs if and only if $-1$ is a
square in the residue field $\mathbb{F}_{q}$. Since $\mathbb{F}_{q}^{\times}$
is a cyclic group of order $q-1$, we infer directly that $-1$ is a square if
and only if $q\equiv 1\mod 4$.

Finally, note that when $p\neq 2$, we choose a set of representatives for
$F^{\times}/\\{\pm 1\\}$ via a choice of representatives of
$\kappa^{\times}/\\{\pm 1\\}$. Let $S_{\kappa}$ be such a set; then via the
isomorphism $\kappa^{\times}\cong\mu_{q-1}\subset\mathcal{R}^{\times}$ of
Example 3.3 we can lift this to a subset of $\mathcal{R}^{\times}$ contained
in the subgroup $\mu_{q-1}$. Then using (3.1) we may take

$F^{\times}/\\{\pm 1\\}=\\{su_{1}\varpi^{n}\mid s\in S_{\kappa},u_{1}\in
1+\mathcal{P},n\in\mathbb{Z}\\}\cong(\mu_{p-1}/\mu_{2})\times(1+\mathcal{P})\times\mathbb{Z}.$

Putting these together gives the following satisfyingly explicit and simple
statement.

######  Theorem 4.4.

Let $F$ be a local nonarchimedean field of odd residual characteristic, and
suppose its residue field $\kappa$ has $q$ elements. Let $K$ be quadratic
extension of $F$. Then the distinct isomorphism classes of nonassociative
quaternion algebras with left nucleus $K$ are given by $(K/F,\sigma,a)$ where
$\sigma\in\mathrm{Gal}(K/F)$ is the nontrivial element and
$a\in\mathcal{S}(K)$, where $\mathcal{S}(K)$ is given as follows:

  1. (1)

If $K=F(\sqrt{\varepsilon})$ is unramified, then

$\mathcal{S}(K)=\\{\sqrt{\varepsilon},\;\varpi\sqrt{\varepsilon},\;1+s\sqrt{\varepsilon},\;\varpi+s\sqrt{\varepsilon}\mid
s\in F^{\times}/\\{\pm 1\\}\\}.$

  2. (2)

If $K=F(\sqrt{\alpha})$ is ramified, where
$\alpha\in\\{\varpi,\varepsilon\varpi\\}$, then

    1. (a)

If If $q\equiv 1\mod 4$ then

$\mathcal{S}(K)=\\{\sqrt{\alpha},\;\varepsilon\sqrt{\alpha},\;1+s\sqrt{\alpha},\;\varepsilon+s\sqrt{\alpha}\mid
s\in F^{\times}/\\{\pm 1\\}\\}.$

    2. (b)

If $q\equiv 3\mod 4$ then

$\mathcal{S}(K)=\\{\sqrt{\alpha},\;\pm 1+s\sqrt{\alpha}\mid s\in
F^{\times}/\\{\pm 1\\}\\}.$

###### Proof.

We apply Theorem 4.2. If $K/F$ is unramified, then
$\mathcal{C}_{K}=\\{1,\varpi\\}$ and $-1\in N_{K/F}(K^{\times})$, yielding the
first case. If it is ramified, then $\mathcal{C}_{K}=\\{1,\varepsilon\\}$. By
the preceding, if $q\equiv 1\mod 4$ then $-1\in(F^{\times})^{2}\subset
N_{K/F}(K^{\times})$, and this yields the second case. In the last case, we
have $-1\notin(F^{\times})^{2}$, which implies we may without loss of
generality choose $\varepsilon=-1$. Thus $\mathcal{C}_{K}=\\{\pm 1\\}$, giving
the desired result. ∎

Note that for $p$-adic fields $F$, $p\not=2$, these algebras lie in class one
or two, or are new examples for class three or four in [Lim2006], since proper
nonassociative quaternion algebras are not isotopic to twisted fields.

###  4.3. Case of $2$-adic fields

The classification of nonassociative quaternion algebras given in Theorem 4.2
applies also to the quadratic extensions of $2$-adic fields (meaning, finite
algebraic extensions of $\mathbb{Q}_{2}$), since these have characteristic
zero. There are two differences.

For one, when $F$ has residual characteristic $2$, we have

$|F^{\times}/(F^{\times})^{2}|=2^{e+2}$

were $e$ is the ramification degree of $F$ over $\mathbb{Q}_{2}$. That is, the
number of distinct quadratic extensions of $F$ increases exponentially with
its absolute ramification index. As these give the distinct choices for the
left nucleus of a nonassociative quaternion algebra, the number of isomorphism
classes of nonassociative quaternion algebras increases exponentially in the
ramification index as well. Nonetheless, for any given choice of $F$, we can
generate an explicit list of representatives of
$\mathcal{R}^{\times}/(\mathcal{R}^{\times})^{2}$ using Hensel’s Lemma 3.2,
and then the distinct quadratic extensions are obtained as $F(\sqrt{c})$ as
$c$ varies over the nontrivial elements of the product

$\\{1,\varpi\\}\times\mathcal{R}^{\times}/(\mathcal{R}^{\times})^{2}.$

For another, while local class field theory implies that
$|\mathcal{C}_{K}|=|F^{\times}/N_{K/F}(K^{\times})|=2$, it can be nontrivial
to general an explicit representative, such as would be obtained from the
Artin reciprocity map. Of course, in degree two this is simply equivalent to
identifying an element of $F^{\times}$ that is not in the image of the norm
map.

To give the flavour of the parametrization of isomorphism classes of
nonassociative quaternion algebras in this case, we present the example of
$F=\mathbb{Q}_{2}$ in detail.

######  Theorem 4.5.

Let $F=\mathbb{Q}_{2}$. Then $F$ has seven distinct quadratic extensions $K$,
enumerated in Table 1. The isomorphism classes of nonassociative quaternion
algebras with left nucleus $K$ are represented by $(K/F,\sigma,a)$ where
$\sigma\in\mathrm{Gal}(K/F)$ is the nontrivial element and
$a\in\mathcal{S}(K)$, where $\mathcal{S}(K)$ is given as follows:

  1. (a)

If $K=\mathbb{Q}_{2}(\sqrt{-3})$, then this extension is unramified and

$\mathcal{S}(K)=\\{\sqrt{-3},\;2\sqrt{-3},\;1+s\sqrt{-3},\;2+s\sqrt{-3}\mid
s\in\mathbb{Q}_{2}^{\times}/\\{\pm 1\\}\\}.$

  2. (b)

If $K=\mathbb{Q}_{2}(\sqrt{\alpha})$ with $\alpha\in\\{2,-6\\}$, then this
extension is ramified and $-1\in N_{K/F}(K^{\times})$. We have

$\mathcal{S}(K)=\\{\sqrt{\alpha},\;3\sqrt{\alpha},\;1+s\sqrt{\alpha},\;3+s\sqrt{\alpha}\mid
s\in\mathbb{Q}_{2}^{\times}/\\{\pm 1\\}\\}.$

  3. (c)

If $K=\mathbb{Q}_{2}(\sqrt{\alpha})$ with $\alpha\in\\{-1,-2,3,6\\}$, then
this extension is ramified and $-1\notin N_{K/F}(K^{\times})$. We have

$\mathcal{S}(K)=\\{\sqrt{\alpha},\;\pm 1+s\sqrt{\alpha}\mid s\in
F^{\times}/\\{\pm 1\\}\\}.$

###### Proof.

We record in Table 1 several facts about the quadratic extensions of
$\mathbb{Q}_{2}$, as follows. Using Hensel’s Lemma, we deduce that there are
no solutions to $x^{2}=\alpha$ for $\alpha\in\\{-1,-3,3\\}$ but that every
element of $1+4\mathbb{Z}_{2}$ is a square; moreover, no element of valuation
equal to $1$ may have a square root in $F$. We deduce that
$\mathbb{Q}_{2}^{\times}/(\mathbb{Q}_{2}^{\times})^{2}=\\{\pm 1,\pm 2,\pm
3,\pm 6\\}$. Thus the first column lists the distinct quadratic extensions of
$\mathbb{Q}_{2}$, with the unramified extension (the one containing the cube
roots of unity) being listed first. The second column is taken from [Shi10,
p34] where the image of the norm map was computed explicitly on a case-by-case
basis. In the third column we record a convenient choice of element $\gamma$
such that $\mathcal{C}_{K}=F^{\times}/N_{K/F}(K^{\times})=\\{1,\gamma\\}$.

The theorem is now a direct application of Theorem 4.2. To specify the cases
of that theorem that apply, we explicitly record in the fourth column of Table
1 whether or not $-1$ is in the image of the norm map, and in the final column
put a label to the corresponding case of the statement of Theorem 4.5.

Table 1. The distinct quadratic extensions of $\mathbb{Q}_{2}$ $K=F(\sqrt{c})$ | $N_{K/F}(K^{\times})/(F^{\times})^{2}$ | $\mathcal{C}_{K}$ |  $-1\in N_{K/F}(K^{\times})$? | Case
---|---|---|---|---
$\mathbb{Q}_{2}(\sqrt{-3})$ | $\\{\pm 1,\pm 3\\}$ | $\\{1,2\\}$ | yes | (a)
$\mathbb{Q}_{2}(\sqrt{-6})$ | $\\{\pm 2,\pm 6\\}$ | $\\{1,3\\}$ | yes | (b)
$\mathbb{Q}_{2}(\sqrt{2})$ | $\\{\pm 1,\pm 2\\}$ | $\\{1,3\\}$ | yes | (b)
$\mathbb{Q}_{2}(\sqrt{-1})$ | $\\{1,-2,-3,6\\}$ | $\\{\pm 1\\}$ | no | (c)
$\mathbb{Q}_{2}(\sqrt{-2})$ | $\\{1,2,3,6\\}$ | $\\{\pm 1\\}$ | no | (c)
$\mathbb{Q}_{2}(\sqrt{3})$ | $\\{1,-2-,3,6\\}$ | $\\{\pm 1\\}$ | no | (c)
$\mathbb{Q}_{2}(\sqrt{6})$ | $\\{1,-2,3,-6\\}$ | $\\{\pm 1\\}$ | no | (c)

∎

###  4.4. Case of Laurent series over a field of characteristic two

Suppose now that we are in the last remaining case, that of characteristic
two.

Note first that extensions of the form $F(\sqrt{c})$ for any
$c\notin(F^{\times})^{2}$ are (purely) inseparable. Instead, the distinct
Galois quadratic extensions over a field $F$ of characteristic $2$ are
obtained, by the Artin–Schreier Theorem [Lan02, Ch VI Thm 6.4, 8.3], as

$K=F(x)/(x^{2}+x+c)$

where $c\in F$ ranges over the nontrivial cosets of the additive subgroup
$\mathfrak{P}_{2}=\\{z^{2}+z\mid z\in F\\}$. Note that if $\alpha\in K$ is a
root of $x^{2}+x+c$ then so is $\alpha+1$, so $K=F(\alpha)$.

In this section, we let $F$ be a local nonarchimedean field of characteristic
two. Then we have $F=\mathbb{F}_{2^{f}}(\\!(t)\\!)$, for some
$f\in\mathbb{N}$.

While $F$ admits a unique quadratic unramified Galois extension, it has
infinitely many distinct ramified Galois quadratic extensions. For example,
since each element of $\mathfrak{P}_{2}\smallsetminus\mathcal{R}$ must have
_even_ valuation, it follows that the elements of
$\\{t^{-2k-1}:k\in\mathbb{N}\\}$, whose distinct differences have odd
valuation, represent infinitely many distinct cosets of $\mathfrak{P}_{2}$.

The classification of the distinct nonassociative quaternion algebras over $F$
correspondingly assumes a different flavour. For one, there will be infinitely
many distinct choices for the left nucleus, corresponding to these various
distinct quadratic Galois extensions. On the other hand, the parametrization
of the isomorphism classes corresponding to a fixed quadratic extension admits
a simpler description, as follows.

######  Theorem 4.6.

Suppose $K=F(\alpha)$ is a separable quadratic extension of a local field $F$
of characteristic $2$. Let $\gamma$ be a nontrivial element of
$F^{\times}/N_{K/F}(K^{\times})$ and $\sigma$ the element of
$\mathrm{Gal}(K/F)$ such that $\sigma(\alpha)=\alpha+1$. The distinct
isomorphism classes of nonassociative quaternion algebras with left nucleus
$K$ are represented by $(K/F,\sigma,a)$ where $a$ is chosen from the set
$\mathcal{S}(K)$ given by

$\mathcal{S}(K)=\\{s+\alpha,\gamma(s+\alpha)\mid s\in F/\\{0,1\\}\\}$

where $F/\\{0,1\\}$ denotes the quotient of the additive group of $F$ by the
two-element prime field $\mathbb{F}_{2}$.

###### Proof.

Let $m(x)=x^{2}+x+c$ be the minimal polynomial of $\alpha$. Its other root is
$\alpha+1$, and these are linearly independent over $F$. Thus

$\\{\alpha,\alpha+1\\}$

forms a basis for $K/F$. Let us evaluate the equivalence relation $\sim$
arising from Theorem 2.6 on $K\smallsetminus F$. Write $a=a_{0}+a_{1}\alpha$;
then $a_{1}\neq 0$ since $a\notin F$. Thus there exists a unique
$r\in\\{1,\gamma\\}$ and $n\in N_{K/F}(K^{\times})$ such that $a_{1}=rn$.
Setting $s=a_{0}n^{-1}$, it follows that $a\in(s+r\alpha)N_{K/F}(K^{\times})$.

Since $\sigma(s+\alpha)=(s+1)+\alpha$ and
$\sigma(s+\gamma\alpha)=(s+\gamma)+\gamma\alpha$, and since the characteristic
of $F$ is $2$, it follows that the distinct isomorphism classes are
parametrized by the set

$\\{s_{1}+\alpha,s_{\gamma}+\gamma\alpha\mid s_{1}\in
F/\\{0,1\\},s_{\gamma}\in F/\\{0,\gamma\\}\\}.$

As the map sending $a\in F$ to $a\gamma\in F$ is an additive group
automorphism sending $\\{0,1\\}$ to $\\{0,\gamma\\}$, we deduce the final form
of the parameter set $\mathcal{S}(K)$. ∎

##  5\. Nonassociative cyclic algebras of degree $m>2$

In this section, we begin with some results for nonassociative cyclic algebras
of degree $m>2$, before specializing to the case that $m$ is prime (and $F$
contains a primitive $m$th root of unity), concluding with an application to
the case of local nonarchimedean fields.

###  5.1. Different generators of $\mathrm{Gal}(K/F)$ give nonisomorphic
algebras

Let $F$ be an arbitrary field such that $F$ admits a cyclic Galois extension
$K$ of degree $m>2$. By Lemma 2.5, the choice of field $K$ is an invariant of
the isomorphism class.

In the case of _associative_ cyclic algebras, that is, when $a\in F^{\times}$,
one has $(K/F,\sigma,a)\cong(K/F,\sigma^{k},a^{k})$ [Pie82, 15.1 Corollary a],
that is, different choices of generator of the Galois group yield isomorphic
algebras.

We show that, in stark contrast, nonassociative cyclic algebras corresponding
to distinct choices of generator $\sigma\in\mathrm{Gal}(K/F)$ are _never_
isomorphic.

######  Theorem 5.1.

Let $F$ be an arbitrary field and let $m\geq 3$ be the degree of a cyclic
Galois extension $K/F$. For any two distinct generators
$\sigma_{1}\neq\sigma_{2}$ of the Galois group $\mathrm{Gal}(K/F)$, and for
any $a_{1},a_{2}\in K\smallsetminus F$, we have

$(K/F,\sigma_{1},a_{1})\not\cong(K/F,\sigma_{2},a_{2}).$

###### Proof.

Suppose to the contrary that there exists an isomorphism
$\varphi\colon(K/F,\sigma_{1},a_{1})\to(K/F,\sigma_{2},a_{2})$; this is an
isomorphism of $F$-vector spaces satisfying $\varphi(uv)=\varphi(u)\varphi(v)$
for all $u,v\in(K/F,\sigma_{1},a_{1})$. In particular, $\varphi$ restricts to
a field automorphism of $K$; since the Galois group is cyclic, and thus
abelian, this implies $\varphi$ commutes with $\sigma_{i}$ on $K$, for
$i\in\\{1,2\\}$.

By Definition 2.1, $(K/F,\sigma_{i},a_{i})=\bigoplus_{s=0}^{m-1}Kt_{i}^{s}$
with an $F$-bilinear multiplication defined for $0\leq s,s^{\prime}<m$ and
$k,k^{\prime}\in K$ by

$(kt_{i}^{s})(k^{\prime}t_{i}^{s^{\prime}})=\begin{cases}k\sigma_{i}^{s}(k^{\prime})t_{i}^{s+s^{\prime}}&\text{if
$s+s^{\prime}<m$;}\\\
k\sigma_{i}^{s}(k^{\prime})a_{i}t_{i}^{s+s^{\prime}-m}&\text{if
$s+s^{\prime}\geq m$.}\end{cases}$

In particular, we infer that the left $K$-submodule $Kt_{i}^{s}$ can be
characterized as

$Kt_{i}^{s}=\\{u\in(K/F,\sigma_{i},a_{i})\mid\forall k\in
K,uk=\sigma_{i}^{s}(k)u\\}.$

Since $\sigma_{1},\sigma_{2}$ both generate $\mathrm{Gal}(K/F)$, there is some
$1<j<m$, with $(j,m)=1$, such that $\sigma_{1}=\sigma_{2}^{j}$. It follows
that $\varphi(t_{1})=kt_{2}^{j}$ for some $k\in K$.

Let $\ell$ be the least positive integer such that $\ell j>m$; since
$\sigma_{1}\neq\sigma_{2}$, we have $\ell<m$. For any index $s\geq 1$, set
$\gamma_{s}(k)=k\sigma_{2}^{j}(k)\sigma_{2}^{2j}(k)\cdots\sigma_{2}^{(s-1)j}(k)$.
Then one can show by induction that

$\varphi(t^{s})=\begin{cases}\gamma_{s}(k)t_{2}^{sj}&\text{if $1\leq
s<\ell$;}\\\ \gamma_{\ell}(k)a_{2}t_{2}^{\ell j-m}&\text{if
$s=\ell$.}\end{cases}$

Since $1\leq\ell<m$, we have

$t_{1}\cdot t_{1}^{\ell}=t_{1}^{\ell}\cdot
t_{1}=\begin{cases}t_{1}^{\ell+1}&\text{if $\ell+1<m$};\\\ a_{1}&\text{if
$\ell+1=m$.}\end{cases}$

However, we have

$\varphi(t_{1}^{\ell})\varphi(t_{1})=\gamma_{\ell}(k)a_{2}t_{2}^{\ell
j-m}\cdot kt_{2}^{j}=\begin{cases}\gamma_{\ell+1}(k)a_{2}t_{2}^{\ell
j+j-m}&\text{if $(\ell+1)j<2m$;}\\\ \gamma_{\ell+1}(k)a_{2}^{2}t_{2}^{\ell
j+j-2m}&\text{if $2m\leq(\ell+1)j$}\end{cases}$

whereas

$\varphi(t_{1})\varphi(t_{1}^{\ell})=kt_{2}^{j}\cdot\gamma_{\ell}(k)a_{2}t_{2}^{\ell
j-m}=\begin{cases}\gamma_{\ell+1}(k)\sigma_{2}^{j}(a_{2})t_{2}^{\ell
j+j-m}&\text{if $(\ell+1)j<2m$;}\\\
\gamma_{\ell+1}(k)\sigma_{2}^{j}(a_{2})a_{2}t_{2}^{\ell j+j-2m}&\text{if
$2m\leq(\ell+1)j$}.\end{cases}$

It follows that $\varphi$ is well-defined only if
$\sigma_{1}(a_{2})=\sigma_{2}^{j}(a_{2})=a_{2}$, meaning that $a_{2}$ lies in
the fixed field of $\sigma_{1}$. But this is impossible: $\sigma_{1}$
generates the cyclic Galois group, and $a_{2}\notin F$, by construction. ∎

###  5.2. An explicit parametrization of isomorphism classes

Now suppose that $m$ is an odd prime and that $F$ contains a primitive $m$th
root of unity. If $F$ has prime characteristic $p$ then this condition implies
that $gcd(m,p)=1$. Kummer theory [Lan02, Ch VI Thm 6.2, 8.2] ensures that the
Galois extensions of $F$ of degree $m$ are in bijection with the subgroups of
$F^{\times}/(F^{\times})^{m}$ of order $m$.

By Theorem 5.1, we may fix a choice of cyclic field extension $K/F$ of degree
$m$ and a choice of generator $\sigma$ of $\mathrm{Gal}(K/F)$. We wish to
classify the algebras $(K/F,\sigma,a)$, as $a$ runs over $K\smallsetminus F$,
up to isomorphism.

Let $\zeta\in F$ be a primitive $m$th root of unity. By Kummer theory, $K$ is
the splitting field of some polynomial $x^{m}-b$, with
$b\notin(F^{\times})^{m}$. Thus we may choose $\beta\in K$ to be a root of
$x^{m}-b$ for which $\sigma(\beta)=\zeta\beta$. Then

$\\{1,\beta,\beta^{2},\ldots,\beta^{m-1}\\}$

is an $F$-basis for $K$ with the property that
$\sigma(\beta^{k})=\zeta^{k}\beta^{k}$ for all $0\leq k<m$.

We first prove that the isomorphism classes of cyclic algebras
$(K/F,\sigma,a)$ admit a coarse partition into $2^{m-1}$ subsets with respect
to this choice of basis, inspired by the classification of quaternion algebras
in Theorem 4.2.

######  Definition 5.2.

Let
$\mathcal{I}=\mathcal{P}(\\{0,1,\ldots,m-1\\})\smallsetminus\\{\\{0\\},\emptyset\\}$
be the set of nonempty subsets of $\\{0,1,\ldots,m-1\\}$, excluding the set
$\\{0\\}$. For each $I\in\mathcal{I}$, define

$K(I)=\\{\sum_{i=0}^{m-1}a_{i}\beta^{i}\mid\forall i\in I,a_{i}\in
F^{\times},\forall i\notin I,a_{i}=0\\}\subset K\smallsetminus F.$

Thus for example if $m=3$, $K(\\{1,2\\})=\\{a_{1}\beta+a_{2}\beta^{2}\mid
a_{1},a_{2}\in F^{\times}\\}$. It follows directly that

$K\smallsetminus F=\bigcup_{I\in\mathcal{I}}K(I),$

and that these sets $K(I)$ are invariant both under the action of $\sigma$,
and under multiplication by elements of $F^{\times}$. Applying Theorem 2.6
gives the following lemma.

######  Lemma 5.3.

Suppose $I,J\in\mathcal{I}$ are distinct. Then for any $a\in K(I)$, $b\in
K(J)$, we have

$(K/F,\sigma,a)\not\cong(K/F,\sigma,b).$

To further refine these partitions, we require the following definition.

######  Definition 5.4.

Let $I\in\mathcal{I}$ be such that $|I|=k+1\geq 2$. Write $F^{[\times k]}$ for
the $k$-fold direct product $F^{\times}\times F^{\times}\times\cdots\times
F^{\times}$. If the elements of $I$ are $i_{0}<i_{1}<\cdots<i_{k}$ then let

$\Delta_{I}=\\{(\zeta^{s(i_{1}-i_{0})},\zeta^{s(i_{2}-i_{0})},\ldots,\zeta^{s(i_{k}-i_{0})})\in
F^{[\times k]}:1\leq s\leq m\\}$

where $\zeta$ is any primitive $m$th root of unity. Write $F^{[\times
k]}/\Delta_{I}$ for any fixed choice of representatives for these cosets. For
any fixed choice of $a_{i_{0}}\in F^{\times}$, define

$K(I;a_{i_{0}})=\\{a_{i_{0}}\beta^{i_{0}}+\sum_{j=1}^{k}a_{i_{j}}\beta^{i_{j}}:(a_{i_{1}},a_{i_{2}},\ldots,a_{i_{k}})\in
F^{[\times k]}/\Delta_{I}\\}.$

Finally, set $K(I;a_{i_{0}})=\\{a_{i_{0}}\beta^{i_{0}}\\}$ when
$I=\\{i_{0}\\}\in\mathcal{I}$.

We may now state our main classification theorem for cyclic extensions of odd
prime degree.

######  Theorem 5.5.

Suppose $m$ is an odd prime and $F$ contains a primitive $m$th root of unity
$\zeta$. Then the distinct isomorphism classes of nonassociative cyclic
algebras of degree $m$ over $F$ are represented by

$(K/F,\sigma,a)$

where:

  * •

$K$ is one of the distinct cyclic Galois field extensions of $F$ of degree
$m$, and $\mathcal{C}_{K}$ denotes a set of representatives of
$F^{\times}/N_{K/F}(K^{\times})$;

  * •

$\sigma$ is a generator of $\mathrm{Gal}(K/F)$; and

  * •

$a\in\mathcal{S}(K)\subset K\smallsetminus F$

where $\mathcal{S}(K)$ is defined as follows.

If $\zeta\in N_{K/F}(K^{\times})$, then we take

$\mathcal{S}(K)=\bigcup_{I\in\mathcal{I},a_{i_{0}}\in\mathcal{C}_{K}}K(I;a_{i_{0}}).$

whereas otherwise, we may take $\mathcal{C}_{K}=\mu_{m}$ and then

$\mathcal{S}(K)=\bigcup_{I\in\mathcal{I}:i_{0}=0,a_{0}\in\mu_{m}}K(I;a_{0})\;\cup\;\bigcup_{I\in\mathcal{I}:i_{0}>0}\\{1+\sum_{j=1}^{|I|-1}a_{i_{j}}\beta^{i_{j}}:\forall
j,a_{i_{j}}\in F^{\times}\\}.$

###### Proof.

By Theorem 5.1 and Lemma 5.3, it suffices to partition $K(I)$ into equivalence
classes under $\sim$ for each fixed $K/F$, $\sigma$ and $I\subset\mathcal{I}$.

Let $\mathcal{C}_{K}$ be a fixed set of representatives for
$F^{\times}/N_{K/F}(K^{\times})$. When $\zeta\notin N_{K/F}(K^{\times})$, it
follows that $\mu_{m}\cap N_{K/F}(K^{\times})=\\{1\\}$ and thus this
$m$-element group must represent the quotient, allowing us to set
$\mathcal{C}_{K}=\mu_{m}$.

Let $a\in K(I)$ and let $i_{0}<i_{1}<\cdots<i_{k}$ be the distinct elements of
$I$. Then $a_{i_{0}}=cn$ for some unique $c\in\mathcal{C}_{K}$ and some $n\in
N_{K/F}(K^{\times})$. Consequently, scaling $a$ by $n^{-1}$ gives

$a\sim c\beta^{i_{0}}+\sum_{j=1}^{k}a_{i_{j}}\beta^{i_{j}}$

for some $a_{i_{j}}\in F^{\times}$ and $c\in\mathcal{C}_{K}$. Furthermore, for
any $s\in\mathbb{Z}$, we have

(5.1)
$\sigma^{s}\left(c\beta^{i_{0}}+\sum_{j=1}^{k}a_{i_{j}}\beta^{i_{j}}\right)=c\zeta^{si_{0}}\beta^{i_{0}}+\sum_{j=1}^{k}a_{i_{j}}\zeta^{si_{j}}\beta^{i_{j}}.$

If $\zeta\in N_{K/F}(K^{\times})$, then we may scale this expression by
$\zeta^{-si_{0}}\in N_{K/F}(K^{\times})$ to produce an equivalent element of
$K(I)$. It follows that the equivalence classes are parametrized by

$\\{c\beta^{i_{0}}+\sum_{j=1}^{k}a_{i_{j}}\beta^{i_{j}}:(a_{i_{1}},\ldots,a_{i_{k}})\in
F^{[\times k]}/\Delta_{I}\\}=K(I;c),$

as $c$ varies over $\mathcal{C}_{K}$, as required.

If instead we have $\mathcal{C}_{K}=\mu_{m}$, then there are two cases. If
$i_{0}=0$, then the relations induced by (5.1) imply directly, as above, that
every element $a\in K(I)$ is equivalent to a unique element of the set
$K(I;c)$. However, if $i_{0}>0$, then as $s$ varies over $\\{1,2,\cdots,m\\}$,
the coefficient of $\beta^{i_{0}}$ in (5.1) varies over $\mathcal{C}_{K}$.
Consequently, every $a\in K(I)$ is equivalent to one for which the first
nonzero coefficient $c$ is equal to $1$, and no two distinct elements of this
form will be equivalent. Taken together, this gives the set $\mathcal{S}(K)$
in this second case. ∎

###  5.3. Application to local nonarchimedean fields

From now on, let $F$ be again a local nonarchimedean field of residual
characteristic $p$ and residue field of order $q$. Let $m$ be an odd prime
distinct from $p$; then all extensions of degree $m$ of $F$ are tamely
ramified. Let us furthermore assume that $F$ contains a primitive $m$th root
of unity, or equivalently, that $m|(q-1)$, where $q=|\kappa|$ is the order of
the residue field of $F$.

Since
$F^{\times}/(F^{\times})^{m}\cong\mathbb{Z}/m\mathbb{Z}\times\mathbb{Z}/m\mathbb{Z}$,
Kummer theory implies that we have precisely $m+1$ distinct Galois extensions
of degree $m$, corresponding to all order-$m$ subgroups, and consequently
$m+1$ collections of isomorphism classes, one for each such extension.

We now make the parametrization given in Theorem 5.5 wholly explicit in this
setting, by defining a set $\mathcal{C}_{K}$ of representatives of
$F^{\times}/N_{K/F}(K^{\times})$ for each degree $m$ extension of $F$, and
determining the conditions under which $\mu_{m}\subset N_{K/F}(K^{\times})$.

######  Proposition 5.6.

Let $K$ be a cyclic Galois field extension of degree $m$ of a local
nonarchimedean field $F$, such that $\mu_{m}\subset F$ and $m\neq p$. Let $q$
denote the order of the residue field $\kappa$ of $F$. Then
$F^{\times}/N_{K/F}(K^{\times})$ is represented by

$\mathcal{C}_{K}=\begin{cases}\\{1,\varpi,\varpi^{2},\cdots,\varpi^{m-1}\\}&\text{if
$K/F$ is unramified;}\\\ \mu_{m}(F)&\text{if $K/F$ is ramified and
$m^{2}\nmid(q-1)$;}\\\ \\{1,\xi,\xi^{2},\cdots,\xi^{m-1}\\}&\text{if $K/F$ is
ramified and $m^{2}|(q-1)$},\end{cases}$

where $\xi\in\mu_{q-1}(F)$ is any choice of primitive $(q-1)$th root of unity
in $F$.

###### Proof.

Since $m\neq p$, Hensel’s Lemma implies that the map $x\mapsto x^{m}$ is a
bijection on $1+\mathcal{P}$, and that $\mu_{q-1}(F)\cong\kappa^{\times}$ as
in Example 3.3.

Thus via the map sending a pair
$(\overline{a},\ell)\in\kappa^{\times}\times\mathbb{Z}$ to $a\varpi^{\ell}$,
we have
$\kappa^{\times}/(\kappa^{\times})^{m}\times\mathbb{Z}/m\mathbb{Z}\cong
F^{\times}/(F^{\times})^{m}$. Now $(F^{\times})^{m}\subset
N_{K/F}(K^{\times})\subset F^{\times}$ and
$|F^{\times}/N_{K/F}(K^{\times})|=m$ by local class field theory. When $K/F$
is unramified, the norm map surjects onto $\mathcal{R}^{\times}$, so we may
take our representatives $\mathcal{C}_{K}$ from powers of $\varpi$. When $K/F$
is (totally) ramified, we instead have
$N_{K/F}(K^{\times})\cap\mathcal{R}^{\times}=(\mathcal{R}^{\times})^{m}$ so is
suffices to identify a set of representatives from among the Teichmüller lifts
of elements of $\kappa^{\times}/(\kappa^{\times})^{m}$.

If $m^{2}\nmid(q-1)$, then no primitive $m$th root of unity can be an $m$th
power, and thus we may take $\mathcal{C}_{K}=\mu_{m}$ as a preferred set of
representatives. Otherwise, one may take the first $m$ powers of any primitive
element of $\mu_{q-1}(F)\cong\kappa^{\times}$ (or indeed, of any element whose
order $k$ satisfies $mk\nmid(q-1)$). ∎

###  5.4. An explicit example: nonassociative cyclic algebras of degree $3$

In this section, we specialize the results of the preceding section to degree
$3$, to better illustrate the combinatorial and explicit nature of the
parametrization.

######  Theorem 5.7.

Let $F$ be a local nonarchimedean field of residual characteristic different
from $3$ such that $F$ contains a primitive cube root of unity $\zeta$. Write
$\mu_{3}=\langle\zeta\rangle$ and set $\Delta_{3}=\\{(a,a^{-1})\in
F^{\times}\times F^{\times}|a\in\mu_{3}\\}$. Then the distinct isomorphism
classes of nonassociative cyclic algebras of degree $3$ over $F$ are
represented by $(K/F,\sigma,a)$ where $\sigma$ is a nontrivial element of
$\mathrm{Gal}(K/F)$ and $K$ and $a\in\mathcal{S}(K)$ are determined as
follows:

  1. (a)

$K=L_{3}$ is the unique unramified extension of $F$ and

$\displaystyle\mathcal{S}(K)=\\{r\beta,$ $\displaystyle
r\beta^{2},r\beta+s\beta^{2},r+s\beta,r+s\beta^{2},r+s_{1}\beta+s_{2}\beta^{2}\;|$
$\displaystyle r\in\\{1,\varpi,\varpi^{2}\\},s\in
F^{\times}/\mu_{3},(s_{1},s_{2})\in(F^{\times}\times
F^{\times})/\Delta_{3}\\}.$

  2. (b)

$K=F(\beta)$ is one of the three ramified extensions, where
$\beta^{3}=u\varpi$ for some
$u\in\mathcal{R}^{\times}/(\mathcal{R}^{\times})^{3}$, and

    1. (a)

if $q\not\equiv 1\mod 9$, then

$\mathcal{S}(K)=\\{r+s_{1}\beta+s_{2}\beta^{2},\beta+s\beta^{2},\beta^{2}|r\in\mu_{3},s\in
F,(s_{1},s_{2})\in(F^{\times}\times F^{\times})/\Delta_{3}\\};$

    2. (b)

if $q\equiv 1\mod 9$, then let
$F^{\times}/N_{K/F}(K^{\times})=\mathcal{R}^{\times}/(\mathcal{R}^{\times})^{3}=\\{1,\gamma,\gamma^{2}\\}$
and set

$\displaystyle\mathcal{S}(K)=\\{r\beta,$ $\displaystyle
r\beta^{2},r\beta+s\beta^{2},r+s\beta,r+s\beta^{2},r+s_{1}\beta+s_{2}\beta^{2}\;|$
$\displaystyle r\in\\{1,\gamma,\gamma^{2}\\},s\in
F^{\times}/\mu_{3},(s_{1},s_{2})\in(F^{\times}\times
F^{\times})/\Delta_{3}\\}.$

###### Proof.

Note that $\mu_{3}\subset\mathcal{R}^{\times}$ implies that $3$ divides $q-1$
or $q\equiv 1\mod 3$, where $q$ is the order of the residue field of $F$.
Similarly, $\zeta$ is itself a cube if and only if $F^{\times}$ contains a
primitive $9$th root of unity, which is the condition that $9|(q-1)$. Here,
$\mathcal{I}=\\{\\{1\\},\\{2\\},\\{0,1\\},\\{1,2\\},\\{0,2\\},\\{0,1,2\\}\\}$.
When $I=\\{i\\}\in\mathcal{I}$, we have $K(I,r)=\\{r\beta^{i}\\}$; when
$I=\\{i_{0}<i_{1}\\}\in\mathcal{I}$, we have
$K(I,r)=\\{r\beta^{i_{0}}+s\beta^{i_{1}}:s\in F^{\times}/\mu_{3}\\}$; and when
$I=\\{0,1,2\\}$, we have
$K(I,r)=\\{r+s_{1}\beta+s_{2}\beta^{2}:(s_{1},s_{2})\in\Delta_{\\{0,1,2\\}}=\Delta_{3}\\}.$
The statement now from Theorem 5.5. ∎

##  6\. Nonassociative cyclic algebras of degree four

When the degree of the nonassociative cyclic algebra is not prime, its
structure becomes more interesting. Let $K/F$ be a cyclic Galois field
extension of degree $m$ with Galois group $G={\rm
Gal}(K/F)=\langle\sigma\rangle$ and $a\in K\smallsetminus F$.

When $m$ is not prime, then there is a nonassociative subalgebra of
$(K/F,\sigma,a)$ associated to each $1<s<m$ such that $(s,m)\neq 1$. Namely,
set $E={\rm Fix}(\sigma^{s})$ to be the subfield of $K$ fixed by $\sigma^{s}$;
then $(K/E,\sigma^{s},a)$ is a cyclic algebra over $E$, that is nonassociative
if $a\notin E$, and that is an $F$-subalgebra of $(K/F,\sigma,a)$.

In particular, if we let $H=\\{\tau\in G\,|\,\tau(a)=a\\}$ be the subgroup of
the (cyclic) Galois group fixing $a$, and let $E={\rm Fix}(H)$, then
$H=\sigma^{s}$ for some divisor $s$ of $m$ and by [Pum2024, Theorem 1]we have

${\rm Nuc}_{r}((K/F,\sigma,a))=(K/E,\sigma^{s},a),$

where this latter is now an associative cyclic algebra of degree $r$ over
$E={\rm Fix}(\sigma^{s})$ (and hence independent of the choice of generator
$\sigma^{s}$).

We briefly discuss the degree four case.

We first consider $F$ an arbitrary field of characteristic different from $2$.
Let $K/F$ be a cyclic Galois field extension of degree four with ${\rm
Gal}(K/F)=\langle\sigma\rangle$. Then $K$ has exactly one quadratic subfield
$E={\rm Fix}(\sigma^{2})$.

Let $a\in K\smallsetminus F$ and consider $(K/F,\sigma,a)$. This is a
16-dimensional algebra over $F$. We know by Theorem 2.4 that if $a\notin E$
then $(K/F,\sigma,a)$ is a division algebra.

Using Definition 2.1 or the general results mentioned above, the
nonassociative cyclic $E$-algebra $B=(K/E,\sigma^{2},a)$ can be embedded as
the $F$-subalgebra generated by $1$ and $t^{2}$ in $(K/F,\sigma,a)$; it is
$8$-dimensional over $F$.

######  Lemma 6.1.

  1. (a)

If $a\not\in E$, then $B$ is a proper nonassociative quaternion division
algebra over $E$, hence also a nonassociative division algebra over $F$.

  2. (b)

If $a\in E$, then $B$ is the right nucleus of $A$. In this case, it is an
associative quaternion algebra over $E$, which is a division algebra if and
only if $a\not\in N_{K/F}(K^{\times})$. Furthermore, then $B$ is a division
algebra if and only if $(K/F,\sigma,a)$ is a division algebra.

We give an explicit, but coarse, classification of these algebras in the case
of local nonarchimedean fields as follows (note that these may not all be
division algebras).

######  Proposition 6.2.

Let $F$ be a local nonarchimedean field of residual characteristic different
from $2$. Then every nonassociative cyclic algebra of degree four over $F$ is
of one of the following types, and moreover, algebras corresponding to
distinct field extensions and distinct generators of the Galois group are
nonisomorphic:

  1. (1)

$(L_{4}/F,\sigma,a)$, corresponding to the unique unramified extension of
degree $4$. Here $B=(L_{4}/F(\sqrt{\varepsilon}),\sigma^{2},a)$.

  2. (2)

$(K_{0}/F,\sigma,a)$, corresponding to the extension
$K_{0}=E(\sqrt{\varepsilon_{E}\varpi})$ where $E=F(\sqrt{\varepsilon})$ and
$\varepsilon_{E}$ is a nonsquare of
$\mathcal{R}_{E}^{\times}\smallsetminus\mathcal{R}^{\times}$. Here
$B=(K_{0}/E,\sigma^{2},a)$.

  3. (3)

If $-1\in(F^{\times})^{2}$: for $i\in\\{1,2,3,4\\}$, $(K_{i}/F,\sigma,a)$,
corresponding to the extension $K_{i}=F(\sqrt[4]{\varepsilon^{i}\varpi})$. The
intermediate subfield is $E=F(\sqrt{\varpi})$ if $i$ is even and
$E=F(\sqrt{\varepsilon\varpi})$ if $i$ is odd. Here
$B=(K_{i}/E,\sigma^{2},a)$.

###### Proof.

Let $A$ be a nonassociative cyclic algebra of degree four over $F$. Since the
left nucleus of $A$ is invariant under isomorphism, it suffices to generate a
list of distinct cyclic Galois extensions $K$ of $F$ of degree $4$. Moreover,
by Theorem 5.1, we know that for the two distinct choices of generator
$\sigma$ of $\mathrm{Gal}(K/F)$ the resulting algebras will be nonisomorphic.

It can be shown that $F$ admits $6$ cyclic Galois extensions of degree four if
$-1\in(F^{\times})^{2}$ and only two otherwise. The first of these is the
unique unramified extension $L_{4}$, with intermediate subfield $E=L_{2}$. The
second is a partially ramified extension $K_{0}$, obtained as the quadratic
extension of $E=L_{2}$ by an element of the form $u\varpi$, where $u$ is
chosen as a nonsquare of $\mathcal{R}_{L_{2}}^{\times}$ that does not lie in
$\mathcal{R}^{\times}$.

When $-1\in(F^{\times})^{2}$, then $F$ contains a primitive $4$th root of
unity, and there are correspondingly four additional Galois totally ramified
extensions, given as $K_{i}=F(\sqrt[4]{\varepsilon^{i}\varpi})$. Their unique
intermediate subfield is $E=F(\sqrt{\varpi})$ if $i$ is even and
$E=F(\sqrt{\varepsilon\varpi})$ if $i$ is odd. ∎

Thus we count four different types for each such field $F$ if
$-1\not\in(F^{\times})^{2}$ and twelve if $-1\in(F^{\times})^{2}$.

While in the associative case, all cyclic algebras of degree $m$ over $F$ are
isomorphic to an algebra of the type $(L_{m}/F,\sigma,a)$ for some $a\in
F^{\times}$, this proposition shows once again that the situation in the
nonassociative setting is more varied.

_Acknowledgements:_ This paper was written while the second author was a
visitor at the University of Ottawa in the spring of 2024. She would like to
thank the Department of Mathematics and Statistics for its hospitality and
congenial atmosphere.

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Vlad-Titus Spătaru primarymatsc200011A25 primarymatsc200011N37 6 2023 96

# Runs of Consecutive Integers Having the Same Number of Divisors

Vlad-Titus Spătaru

###### Abstract.

Our objective is to provide an upper bound for the length $\ell_{N}$ of the
longest run of consecutive integers smaller than $N$ which have the same
number of divisors. We prove in an elementary way that $\log\ell_{N}\ll(\log
N\log\log N)^{\lambda}$, where $\lambda=1/2$. Using estimates for the
Jacobsthal function, we then improve the result to $\lambda=1/3$.

###### keywords:

divisor counting function

###### keywords:

consecutive equidivisible integers

##  1\. Introduction

Let $d(n)$ denote the number of positive divisors of $n$. The equation
$d(n)=d(n+k)$ has been studied extensively. Spiro [Spi81] showed that it has
infinitely many solutions for $k=5040$. Subsequently, Heath-Brown [HB84]
established the case $k=1$, and Pinner [Pin97] ultimately proved that all
values of $k$ yield infinitely many solutions.

As $d(n)$ is equal to $d(n+1)$ infinitely often, one naturally wonders _how
many_ consecutive integers can there be, having the same number of divisors.
Erdős and Mirsky [EM52] conjectured that there are arbitrarily long such runs
of integers. They were not able to provide any estimates for the length of
such sequences: “A related problem consists in the estimation of the longest
run of consecutive integers $\leqslant x$ all of which have the same number of
divisors. This problem seems to be one of exceptional difficulty, and we
[Erdős & Mirsky] have not been able to make any progress with it.”

Our principal objective is to provide an upper bound for the length $\ell_{N}$
of the runs in question. In Section 2, we estimate the order of magnitude of
the $d(n)$ and $\omega(n)$ functions and obtain the following estimate in an
elementary manner:

######  Theorem 1.

Let $\ell_{N}$ denote the length of the longest run of consecutive integers
smaller than $N$, having the same number of divisors. Then,

$\log\ell_{N}\ll\sqrt{\log N\cdot\log\log N}.$

Subsequently, in Section 3, we provide a quicker proof of Theorem 1 based on
the Prime Number Theorem. We also get the following explicit form of Theorem
1:

$\log\ell_{N}\leqslant\sqrt{(1/2+o(1))\log N\log\log N}.$

Finally, in Section 4 we prove a stronger version of Theorem 1 using estimates
of the Jacobsthal function deduced from Brun’s sieve method. We obtain the
following result:

######  Theorem 2.

Let $\ell_{N}$ denote the length of the longest run of consecutive integers
smaller than $N$, having the same number of divisors. Then,

$\log\ell_{N}\ll\sqrt[3]{\log N\cdot\log\log N}.$

##  2\. An Elementary Proof of Theorem 1

In proving Theorem 1, we will make use of the following lemmas, the first
being proven in an elementary manner in [Far09] and the second being Mertens’
bound.

######  Lemma 1.

Let $n$ be a positive integer. Then, $\lcm(1,2,\ldots,n+1)\geqslant 2^{n}$.

######  Lemma 2.

The sum of the reciprocals of the prime numbers not exceeding $n$ satisfies

$\sum_{p\leqslant n}\frac{1}{p}=\log\log n+M+O\left(\frac{1}{\log
n}\right)\ll\log\log n.$

Note that it suffices to prove that Theorem 1 holds for large enough $N$.
Assume that there exist $k>2$ consecutive numbers smaller than $N$, having the
same number of divisors. Let them be $n+1$, $n+2$, …, $n+k$ and write

$d(n+1)=d(n+2)=\cdots=d(n+k)=D.$

We will firstly provide an estimate for $D$, in terms of $k$. For simplicity,
let $K=\lfloor\log_{2}k\rfloor$.

As $k\geqslant 2^{K}$, all residues modulo $2^{K}$ are among $n+1$, $n+2$, …,
$n+k$. Therefore, for all $1\leqslant i\leqslant K-1$, there exists some
$1\leqslant t_{i}\leqslant k$ such that $n+t_{i}\equiv 2^{i}\bmod{2^{K}}$.
Consequently, $\nu_{2}(n+t_{i})=i$, so $i+1$ divides $d(n+t_{i})=D$.

Hence, $D$ is divisible by $\lcm(1,2,\ldots,K)$. Using Lemma 1, we infer that

$D\geqslant\lcm(1,2,\ldots,K)\geqslant 2^{K-1}.$

Recall that $K=\lfloor\log_{2}k\rfloor\geqslant\log_{2}k-1$, so $D\geqslant
k/4$.

Let $\omega(n)$ denote the number of distinct prime factors of $n$. Choose
$1\leqslant l\leqslant k$ arbitrarily. As $n+l\leqslant N$, it follows that
$\nu_{p}(n+l)\leqslant\log_{p}N\leqslant\log_{2}N$ for all prime numbers $p$.
Therefore,

$D=d(n+l)=\prod_{p}(\nu_{p}(n+l)+1)\leqslant\prod_{p\mid
n+l}(\log_{2}N+1)=(\log_{2}N+1)^{\omega(n+l)}.$

Taking logarithms, it follows that $\omega(n+l)\geqslant\log
D/\log(\log_{2}N+1)$. Moreover, note that a prime number $p$ can divide at
most $\lceil k/p\rceil\leqslant k/p+1$ numbers among $n+1$, …, $n+k$.
Therefore, using Lemma 2, it follows that

$\omega((n+1)\cdots(n+k))\geqslant\sum_{i=1}^{k}\omega(n+i)-\sum_{p\leqslant
k}\frac{k}{p}\geqslant\frac{k\log D}{\log(\log_{2}N+1)}-C_{1}k\log\log k,$

for a suitable constant $C_{1}$. Further, we will write
$\log(\log_{2}N+1)\leqslant C_{2}\log\log N$, for some constant $C_{2}$.
Recall that $D\geqslant k/4$, so we have

$\omega((n+1)\cdots(n+k))\geqslant\frac{k\log(k/4)}{C_{2}\log\log
N}-C_{1}k\log\log k.$ (1)

Write the right-hand side of equation 1 as $k\cdot f_{N}(k)$. Clearly, if
$\omega(a)\geqslant b$ then $a\geqslant b!$. Using this remark on equation 1,
we get $(n+1)\cdots(n+k)\geqslant\lceil k\cdot f_{N}(k)\rceil!$. Moreover,
because $N^{k}\geqslant(n+1)\cdots(n+k)$, by applying the well-known
inequality $\log t!\geqslant t\log t-t$, we have

$\displaystyle k\log N$
$\displaystyle\geqslant\log\left((n+1)\cdots(n+k)\right)\geqslant\log\left(\lceil
k\cdot f_{N}(k)\rceil!\right)$ $\displaystyle\geqslant k\cdot
f_{N}(k)\cdot\log(k\cdot f_{N}(k))-k\cdot f_{N}(k).$ (2)

Finally, dividing equation 2 by $k$ we obtain

$\log N\geqslant f_{N}(k)\cdot\log(k\cdot f_{N}(k))-f_{N}(k).$ (3)

Define the interval $I_{N}=\left[\exp\left(C_{1}\cdot C_{2}\cdot\log\log
N\right),\infty\right)$. Using standard arguments, one may infer that $f_{N}$
is increasing on $I_{N}$.

Let us suppose, for the sake of contradiction, that $k>\exp\left(C\sqrt{\log
N\log\log N}\right)$, where $C>\max(\sqrt{C_{2}},C_{1}\cdot C_{2})$. Firstly,
note that since $\log N>\log\log N$ and $C>C_{1}\cdot C_{2}$ then
$\exp\left(C\sqrt{\log N\log\log N}\right)$ and $k$ are in $I_{N}$. Therefore,
we have

$\displaystyle f_{N}(k)$ $\displaystyle>f_{N}\left(\exp\left(C\sqrt{\log
N\log\log N}\right)\right)$ $\displaystyle=\frac{C}{C_{2}}\sqrt{\frac{\log
N}{\log\log N}}-\frac{\log 4}{C_{2}\log\log N}-C_{1}\log\left(C\sqrt{\log
N\log\log N}\right).$ (4)

Viewing equation 2 as a function in $N$, it is evident that for large enough
$N$ (greater than some $N_{1}$) we also have $f_{N}(k)>e$. In what follows, we
will assume that $N>N_{1}$.

As $f_{N}(k)>e$, it follows from equation 3 that $\log N\geqslant
f_{N}(k)\cdot\log k$. Further, applying equation 2 and the estimate for $k$
and isolating the term $\log N$, we get

$\frac{C\log 4}{C_{2}}\sqrt{\frac{\log N}{\log\log N}}+C_{1}C\sqrt{\log
N\log\log N}\log\left(C\sqrt{\log N\log\log
N}\right)\geqslant\left(\frac{C^{2}}{C_{2}}-1\right)\log N.$

Recall that $C>\sqrt{C_{2}}$, so the latter inequality is absurd for large
enough $N$ (greater than some $N_{2}$), as the left-hand side is
asymptotically much smaller than $\log N$. Therefore, Theorem 1 holds for
$N>\max(N_{1},N_{2})$ and $C>\max(\sqrt{C_{2}},C_{1}\cdot C_{2})$.

##  3\. An Explicit Form of Theorem 1

Let $\Omega(n)$ denote the number of prime factors of $n$, counting
multiplicities. Note that $2^{\omega(n)}\leqslant d(n)\leqslant 2^{\Omega(n)}$
for all positive integers $n$. Further, we have the following estimates:

######  Lemma 3.

The product $n\\#$ of the prime numbers not exceeding $n$ satisfies $\log
n\\#\sim n$.

######  Lemma 4.

The sum of $1/\log p$ taken over the prime numbers not exceeding $n$ satisfies

$\sum_{p\leqslant n}\frac{1}{\log p}=\frac{n}{(\log
n)^{2}}+O\left(\frac{n\log\log n}{(\log n)^{3}}\right)=o(1)\cdot\frac{n}{\log
n}.$

Let $M$ be the greatest positive integer satisfying $M\\#\leqslant k$. Then,
there exists $1\leqslant m\leqslant k$ so that $M\\#$ divides $n+m$. Hence,
for any $1\leqslant i\leqslant k$ we have

$2^{\Omega(n+i)}\geqslant d(n+i)=d(n+m)\geqslant d(M\\#)\geqslant
2^{\omega(M\\#)}=2^{\pi(M)},$

so $\Omega(n+i)\geqslant\pi(M)$. It follows that
$\Omega(n+1)+\Omega(n+2)+\cdots+\Omega(n+k)\geqslant k\pi(M)$.

Fix a prime number $p\leqslant k$. Then, for every exponent
$t\leqslant\log_{p}N$ there are at most $\lceil k/p^{t}\rceil\leqslant
k/p^{t}+1$ numbers divisible by $p^{t}$ among $n+1$, $n+2$, …, $n+k$.
Furthermore, if $t>\log_{p}N$, then $p^{t}>N$ so none of $n+1$, $n+2$, …,
$n+k$ are divisible by $p^{t}$.

Following the same double-counting technique as in Legendre’s Theorem, we may
infer

$\sum_{i=1}^{k}\nu_{p}(n+i)\leqslant\sum_{t=1}^{\lfloor\log_{p}N\rfloor}\left(1+\frac{k}{p^{t}}\right)<\frac{\log
N}{\log p}+\frac{k}{p-1}.$ (5)

Now, observe that $n+i$ has at most $\log_{k}N$ prime factors greater than
$k$, including multiplicities. Combining this observation with equation 5 and
Lemmas 2 and 4, we get

$\displaystyle k\pi(M)$
$\displaystyle\leqslant\sum_{i=1}^{k}\Omega(n+i)=\sum_{p>k}\sum_{i=1}^{k}\nu_{p}(n+i)+\sum_{p\leqslant
k}\sum_{i=1}^{k}\nu_{p}(n+i)$ $\displaystyle\leqslant\frac{k\log N}{\log
k}+\sum_{p\leqslant k}\left(\frac{\log N}{\log
p}+\frac{k}{p-1}\right)=(1+o(1))\frac{k\log N}{\log k}+O(k\log\log k).$ (6)

It follows from Lemma 3 and the Prime Number Theorem that $\pi(M)\sim\log
k/\log\log k$. Further, note that $k\log\log k=o(1)\cdot k\log k/\log\log k$.
Using these observations in equation 3 we get

$\frac{k\log k}{\log\log k}\sim k\pi(M)\leqslant(1+o(1))\frac{k\log N}{\log
k}+o(1)\frac{k\log k}{\log\log k}.$

Dividing through $k$ and isolating the remaining functions in $k$ on the left-
hand side, we have $(\log k)^{2}/\log\log k\leqslant(1+o(1))\log N.$ Using
standard arguments, we finally get

$\log k\leqslant\sqrt{(1/2+o(1))\log N\log\log N}.$

##  4\. The Proof of Theorem 2

We will keep the notation used in Section 2. We will require the following
estimates:

######  Lemma 5.

The sum of the prime numbers not exceeding $n$ satisfies

$\sum_{p\leqslant n}p=\frac{n^{2}}{2\log n}+O\left(\frac{n^{2}}{(\log
n)^{2}}\right)\sim\frac{n^{2}}{2\log n}.$

######  Lemma 6.

Let $n$ be a positive integer and $p_{\text{min}}$ be its smallest prime
divisor. Then,

$\frac{\log n}{\log p_{\text{min}}}\geqslant\sum_{p}(p-1)\nu_{p}(d(n)).$

###### Proof.

Throughout the rest of the proof, the letters $p$ and $q$ will refer strictly
to prime numbers. Note that since $p_{\text{min}}$ is the smallest prime
factor of $n$, by taking logarithms we get

$\log n=\sum_{q}\nu_{q}(n)\log q\geqslant\log
p_{\text{min}}\sum_{q}\nu_{q}(n).$ (7)

Further, using the fact that $m^{n}-1\geqslant n(m-1)$ for all positive
integers, we infer that

$k=\prod_{p}p^{\nu_{p}(k+1)}-1\geqslant\sum_{p}\left(p^{\nu_{p}(k+1)}-1\right)\geqslant\sum_{p}(p-1)\nu_{p}(k+1)$
(8)

for any positive integer $k$. Using inequality 8 on $\nu_{q}(n)$ in equation
7, we further have

$\displaystyle\frac{\log n}{\log p_{\text{min}}}$
$\displaystyle\geqslant\sum_{q}\nu_{q}(n)\geqslant\sum_{q}\sum_{p}(p-1)\nu_{p}(\nu_{q}(n)+1)=\sum_{p}\left((p-1)\sum_{q}\nu_{p}(\nu_{q}(n)+1)\right)$
$\displaystyle=\sum_{p}(p-1)\nu_{p}\left(\prod_{q}(\nu_{q}(n)+1)\right)=\sum_{p}(p-1)\nu_{p}(d(n)),$

giving us the desired result. ∎

The proof now hinges on finding an index $i$ for which $n+i$ has a large
minimal prime factor. Jacobsthal [Jac60] defines the function $j(n)$ to be the
least integer so that amongst any $j(n)$ consecutive integers there exists at
least one relatively prime to $n$.

Therefore, if a positive integer $M$ satisfies $j(M\\#)\leqslant k$, then some
$n+i$ has the minimal prime factor larger than $M$. As pointed out by Erdős
[Erd62], it follows directly from Brun’s sieve that $j(n)\ll\omega(n)^{C}$ for
a suitable constant $C$, hence $\log j(n)\ll\log\omega(n)$.

It follows that $\log j(M\\#)\ll\log\omega(M\\#)=\log\pi(M)\ll\log M$, so
among our $k$ consecutive integers we may find one, $n+i$, whose minimal prime
factor $p_{\text{min}}$ satisfies $\log p_{\text{min}}\gg\log k$. Further,
recall that every prime number not exceeding $\log_{2}k$ divides $D=d(n+i)$.
Applying Lemmas 5 and 6 we then get

$\frac{\log N}{\log k}\gg\frac{\log(n+i)}{\log
p_{\text{min}}}\geqslant\sum_{p}(p-1)\nu_{p}(D)\gg\sum_{p\leqslant\log_{2}k}p\sim\frac{(\log_{2}k)^{2}}{2\log\log_{2}k}.$

Consequently, $\log N\gg(\log k)^{3}/\log\log k$, from which Theorem 2 easily
follows.

## Acknowledgments

I would like to express my deep gratitude to Andrew Granville, for his support
and for his contribution [Gra23] with the material presented throughout
Sections 3 and 4. Additionally, I would like to thank Alexandru Gica for his
constant guidance, his insightful comments, and for proofreading my paper.

## References

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  * [Jac60] E. E. Jacobsthal, Über sequenzen ganzer zahlen: von denen keine zu $n$ teilerfremd ist, _Det Kongelige Norske Videnskabers Selskabs Forhandlinger_ 33 (1960), 117–124.
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