Jan Minac, Andrew Schultz, John Swallow
In the mid-1960s Borevic and Faddeev initiated the study of the Galois module structure of groups of pth-power classes of cyclic extensions K/F of pth-power degree. They determined the structure of these modules in the case when F is a local field. In this paper we determine these Galois modules for all base fields F.
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Let p>2 be prime, and let n,m be positive integers. For cyclic field extensions E/F of degree p^n that contain a primitive pth root of unity, we show that the associated F_p[Gal(E/F)]-modules H^m(G_E,mu_p) have a sparse decomposition. When E/F is additionally a subextension of a cyclic, degree p^{n+1} extension E'/F, we give a more refined F_p[Gal(E/F)]-decomposition of H^m(G_E,mu_p).
We establish automatic realizations of Galois groups among groups M\rtimes G, where G is a cyclic group of order p^n for a prime p and M is a quotient of the group ring Fp[G].
We present several constraints on the absolute Galois groups G_F of fields F containing a primitive pth root of unity, using restrictions on the cohomology of index p normal subgroups from a previous paper by three of the authors. We first classify all maximal p-elementary abelian-by-order p quotients of such G_F. In the case p>2, each such quotient contains a unique closed index p elementary abelian subgroup. This seems to be the first case in which one can completely classify nontrivial quotients of absolute Galois groups by characteristic subgroups of normal subgroups. We then derive analog…
Let p be a prime and suppose that K/F is a cyclic extension of degree p^n with group G. Let J be the F_pG-module K^*/K^{*p} of pth-power classes. In our previous paper we established precise conditions for J to contain an indecomposable direct summand of dimension not a power of p. At most one such summand exists, and its dimension must be p^i+1 for some 0<=i<n. We show that for all primes p and all 0<=i<n, there exists a field extension K/F with a summand of dimension p^i+1.
Galois module structure of cyclic extensions of local Fields of characteristic zero
本文描述了特征零局部域循环扩张的精确Galois模结构及通常作为其直接因子的参数化空间Wa,b,m,n。
Let K/F be a cyclic field extension of odd prime degree. We consider Galois embedding problems involving Galois groups with common quotient Gal(K/F) such that corresponding normal subgroups are indecomposable Fp[Gal(K/F)]-modules. For these embedding problems we prove conditions on solvability, formulas for explicit construction, and results on automatic realizability.
Locally cyclic extensions with Galois group $GL_2(p)$
本研究利用椭圆曲线的伽罗瓦表示，构造了所有分解群均为循环群、伽罗瓦群为$GL_2(p)$的有理数域伽罗瓦扩张，首次实现对所有素数p的这类构造。
We realize Frobenius conjugacy classes in Galois groups of certain $q$-polynomials over $\mathbb{F}_q(t)$ using specific degree 1 ideals. We combine this with methods from elementary linear algebra and group theory to realize transvections in some linear Galois groups. This enables the Galois group to be identified as a known classical group in several reasonably general cases.
We present several constraints on the absolute Galois groups G_F of fields F containing a primitive pth root of unity, using restrictions on the cohomology of index p normal subgroups from a previous paper by three of the authors. We first classify all maximal p-elementary abelian-by-order p quotients of such G_F. In the case p>2, each such quotient contains a unique closed index p elementary abelian subgroup. This seems to be the first case in which one can completely classify nontrivial quotients of absolute Galois groups by characteristic subgroups of normal subgroups. We then derive analog…
Let E be a cyclic extension of pth-power degree of a field F of characteristic p. For all m, s in N, we determine K_mE/p^sK_mE as a (Z/p^sZ)[Gal(E/F)]-module. We also provide examples of extensions for which all of the possible nonzero summands in the decomposition are indeed nonzero.
Let p be an odd prime satisfying Vandiver's conjecture. We consider two objects, the Galois group X of the maximal unramified abelian pro-p extension of the compositum of all Z_p-extensions of the pth cyclotomic field and the Galois group G of the unramified pro-p extension of the cyclotomic field of all p-power roots of unity. We give a lower bound for the height of the annihilator of X as an Iwasawa module. Under some mild assumptions on Bernoulli numbers, we provide a necessary and sufficient condition for G to be abelian. The bound and the condition in the two results are given in terms of…
Let $K$ be a local field of characteristic $p$ and let $L/K$ be a totally ramified Galois extension such that Gal$(L/K)\cong C_{p^n}$. In this paper we find sufficient conditions for $L/K$ to admit a Galois scaffold. This leads to sufficient conditions for the ring of integers $O_L$ to be free of rank 1 over its associated order $A_0$, and to stricter conditions which imply that $A_0$ is a Hopf order in the group ring $K[C_{p^n}]$.
Let p be a prime and F a field containing a primitive pth root of unity. Then for n in N, the cohomological dimension of the maximal pro-p-quotient G of the absolute Galois group of F is <=n if and only if the corestriction maps H^n(H,Fp) -> H^n(G,Fp) are surjective for all open subgroups H of index p. Using this result we derive a surprising generalization to dim_Fp H^n(H,Fp) of Schreier's formula for dim_Fp H^1(H,Fp).
Construction of a cyclic $p$-extension of number fields whose unit group has prescribed Galois module structure
本文证明对任意有限循环p-群G，每个无Z_p-挠的有限生成Z_p[G]-模都可在相差自由直和项的意义下实现为某数域G-扩张的单位群，从而刻画了单位群的伽罗瓦模结构。
Let $p$ be prime, and $n,m \in \mathbb{N}$. When $K/F$ is a cyclic extension of degree $p^n$, we determine the $\mathbb{Z}/p^m\mathbb{Z}[\text{Gal}(K/F)]$-module structure of $K^\times/K^{\times p^m}$. With at most one exception, each indecomposable summand is cyclic and free over some quotient group of $\text{Gal}(K/F)$. For fixed values of $m$ and $n$, there are only finitely many possible isomorphism classes for the non-free indecomposable summand. These Galois modules act as parameterizing spaces for solutions to certain inverse Galois problems, and therefore this module computation provid…
Infinitely many elliptic curves over ${\bf Q}$ have a Galois-stable cyclic subgroup of order 4. Such subgroups come in pairs, which intersect in their subgroups of order 2. Let $N_i(X)$ denote the number of elliptic curves over ${\bf Q}$ with at least $i$ pairs of Galois-stable cyclic subgroups of order 4, and height at most $X$. In this article we show that $N_1(X) = c_{1,1}X^{1/3}+c_{1,2}X^{1/6}+O(X^{0.105})$. We also show, as $X\to \infty$, that $N_2(X)=c_{2,1}X^{1/6}+o(X^{1/12})$, the precise nature of the error term being related to the prime number theorem and the zeros of the Riemann ze…
In cyclic, degree 8 extensions of algebraic number fields $N/K$, ambiguous ideals in N are canonical $\mathbb{Z}[C_8]$-modules. Their $\mathbb{Z}[C_8]$-structure is determined here. It is described in terms of indecomposable modules and determined by ramification invariants. Although infinitely many indecomposable $\mathbb{Z}[C_8]$-modules are available (classification by Yakovlev), only 23 appear.
Recently, much work has been done to investigate Galois module structure of local field extensions, particularly through the use of Galois scaffolds. Given a totally ramified $p$-extension of local fields $L/K$, a Galois Scaffold gives us a $K$-basis for $K[G]$ whose effect on the valuation of elements of $L$ is easy to determine. In 2013, N.P. Byott and G.G. Elder gave sufficient conditions for the existence of Galois scaffolds for cyclic extensions of degree $p^2$ in characteristic $p$. We take their work and adapt it to cyclic extensions of degree $p^2$ in characteristic $0$.
Galois module structure of algebraic integers of cyclic cubic fields
研究通过根式通用循环三次多项式确定三次循环域中整数环的伽罗瓦模结构，并显式给出其生成元。
Let p be a prime and F a field containing a primitive pth root of unity. Let E/F be a cyclic extension of degree p and G_E < G_F the associated absolute Galois groups. We determine precise conditions for the cohomology group H^n(E)=H^n(G_E,Fp) to be free or trivial as an Fp[Gal(E/F)]-module. We examine when these properties for H^n(E) are inherited by H^k(E), k>n, and, by analogy with cohomological dimension, we introduce notions of cohomological freeness and cohomological triviality. We give examples of H^n(E) free or trivial for each n in N with prescribed cohomological dimension.
Wild Galois representations: elliptic curves with wild cyclic reduction
本文完成了剩余特征为2或3且曲线在分歧次数可被p整除的扩张上获得良好约化的所有剩余wild情形下椭圆曲线Galois表示的显式计算。
An infinite family of pairs of distinct quartic Galois CM-fields with the same discriminant and regulator
本文构造了无限多个具有相同判别式和调节数的虚二元四次域和虚循环四次域的配对，并给出了具有相同判别式、调节数和类数的示例。
We prove that all imaginary biquadratic fields and cyclic quartic fields of class number $1$ are Euclidean.
We study the inverse Galois problem with local conditions. In particular, we ask whether every finite group occurs as the Galois group of a Galois extension of $\mathbb{Q}$ all of whose decomposition groups are cyclic (resp., abelian). This property is known for all solvable groups due to Shafarevich's solution of the inverse Galois problem for those groups. It is however completely open for nonsolvable groups. In this paper, we provide general criteria to attack such questions via specialization of function field extensions, and in particular give the first infinite families of Galois realiza…
Upon quotienting by units, the elements of norm 1 in a number field $K$ form a countable subset of a torus of dimension $r_1 + r_2 - 1$ where $r_1$ and $r_2$ are the numbers of real and pairs of complex embeddings. When $K$ is Galois with cyclic Galois group we demonstrate that this countable set is equidistributed in a finite cover of this torus with respect to a natural partial ordering induced by Hilbert's Theorem 90.
Let $G$ and $N$ be two finite groups of the same order. It is well-known that the existences of the following are equivalent: (a) a Hopf-Galois structure of type $N$ on any Galois $G$-extension; (b) a skew brace with additive group $N$ and multiplicative group $G$; (c) a regular subgroup isomorphic to $G$ in the holomorph of $N$. We shall say that $(G,N)$ is realizable when any of the above exists. Fixing $N$ to be a cyclic group, W. Rump (2019) has determined the groups $G$ for which $(G,N)$ is realizable. In this paper, fixing $G$ to be a cyclic group instead, we shall give a complete charac…
We present a simple method for the construction of polynomials with cyclic Galois groups, hoping to encourage a reader with some background in algebra to make computations of his/her own.
We show that finite Galois extensions with cyclic Galois group are radical.
We estimate several probability distributions arising from the study of random, monic polynomials of degree $n$ with coefficients in the integers of a general $p$-adic field $K_{\mathfrak{p}}$ having residue field with $q= p^f$ elements. We estimate the distribution of the degrees of irreducible factors of the polynomials, with tight error bounds valid when $q> n^2+n$. We also estimate the distribution of Galois groups of such polynomials, showing that for fixed $n$, almost all Galois groups are cyclic in the limit $q \to \infty$. In particular, we show that the Galois groups are cyclic with p…
This paper is devoted to the explicit description of the Galois descent obstruction for hyperelliptic curves of arbitrary genus whose reduced automorphism group is cyclic of order coprime to the characteristic of their ground field. Along the way, we obtain an arithmetic criterion for the existence of a hyperelliptic descent. The obstruction is described by the so-called arithmetic dihedral invariants of the curves in question. If it vanishes, then the use of these invariants also allows the explicit determination of a model over the field of moduli; if not, then one obtains a hyperelliptic mo…
On The Cyclicity of Algebraic Lattices
本文研究代数格的循环性，通过Minkowski嵌入在伽罗瓦数域中建立循环格的充要条件，并给出理想格循环性的判定条件。
For fields F of characteristic not p containing a primitive $p$th root of unity, we determine the Galois module structure of the group of $p$th-power classes of K for all cyclic extensions K/F of degree p.
Let F be a field containing a primitive pth root of unity, and let U be an open normal subgroup of index p of the absolute Galois group G_F of F. Using the Bloch-Kato Conjecture we determine the structure of the cohomology group H^n(U,Fp) as an Fp[G_F/U]-module for all n in N. Previously this structure was known only for n=1, and until recently the structure even of H^1(U,Fp) was determined only for F a local field, a case settled by Borevic and Faddeev in the 1960s. We apply these results to study partial Euler-Poincare characteristics of open subgroups N of the maximal pro-p quotient T of G_…
We construct a family of quartic polynomials with cyclic Galois group and show that the roots of the polynomials are fundamental units or generate a subgroup of index 5.
Weinberger in 1972, proved that the ring of integers of a number field with unit rank at least $1$ is a principal ideal domain if and only if it is a Euclidean domain, provided the generalised Riemann hypothesis holds. Lenstra extended the notion of Euclidean domains in order to capture Dedekind domains with finite cyclic class group and proved an analogous theorem in this setup. More precisely, he showed that the class group of the ring of integers of a number field with unit rank at least $1$ is cyclic if and only if it has a Euclidean ideal class, provided the generalised Riemann hypothesis…
On cubic polynomials with the cyclic Galois group
本文研究具有循环伽罗瓦群的三次多项式，通过构造二次多项式对根的循环置换引入耦合、相邻类及超类概念，并给出超类的完整描述。
Local Minkowski units in non-abelian extensions with cyclic Sylow $p$-subgroups
本文针对伽罗瓦群为非p群与循环p群直积的伽罗瓦扩张，建立了p处局部Minkowski单位的存在准则，应用该准则构造了满足条件的非阿贝尔扩张，分析了相关Iwasawa模结构并推广了研究范围。
Cup product of inhomogeneous Tate cochains, and Galois cohomology of tori over local fields that split over cyclic extensions
本文给出Tate上同调中杯积的非齐次上链公式，并用于计算分裂于循环扩张的局部域上环面的一阶上同调类的显式上循环。
When is a 2-Power Cyclotomic Extension cyclic?
本文通过Galois群结构、子扩张性质、塔分解及基域特定条件，刻画了2-幂分圆扩张的循环性质。
In our previous paper we describe the Galois module structures of $p$th-power class groups $K^\times/{K^{\times p}}$, where $K/F$ is a cyclic extension of degree $p$ over a field $F$ containing a primitive $p$th root of unity. Our description relies upon arithmetic invariants associated with $K/F$. Here we construct field extensions $K/F$ with prescribed arithmetic invariants, thus completing our classification of Galois modules $K^{\times}/K^{\times p}$.
Let $φ_p(z)=(z-1)^p+2-ζ_p$, where $ζ_p\in\bar{\mathbb{Q}}$ is a primitive $p$-th root of unity for some odd prime $p$. Building on previous work, we show that the $n$-th iterate $φ_p^n(z)$ has Galois group $[C_p]^n$, an iterated wreath product of cyclic groups, whenever $p$ is not a Wieferich prime.
Stabilization on ideal class groups in potential cyclic towers
本文研究数域的势循环$p$塔结构，将岩泽理论中Fukuda的循环$p$塔$p$类群稳定性结果、以及$\u2124_p$扩张的岩泽类数公式，分别推广至势循环$p$塔和势$\u2124_p$扩张情形。
This paper justifies an assertion in (Elder, Proc AMS 137 (2009), no 4, 1193--1203) that Galois scaffolds make the questions of Galois module structure tractable. Let $k$ be a perfect field of characteristic $p$ and let $K=k((T))$. For the class of characteristic $p$ elementary abelian $p$-extensions $L/K$ with Galois scaffolds described in mentioned paper, we give a necessary and sufficient condition for the valuation ring $\mathfrak{O}_L$ to be free over its associated order $\mathcal{A}_{L/K}$ in $K[\Gal(L/K)]$. Interestingly, this condition agrees with the condition found by Y. Miyata, con…
We obtain strong information on the asymptotic behaviour of the counting function for nilpotent Galois extensions with bounded discriminant of arbitrary number fields. This extends previous investigations for the case of abelian groups. In particular, the result confirms a conjecture by the second author on this function for arbitrary groups in the nilpotent case. We further prove compatibility of the conjecture with taking wreath products with the cyclic group of order 2 and give examples in degree up to 8.
Hopf--Galois structures of cyclic type on parallel extensions of prime power degree
本文研究了当扩展L/K的度数为素数幂且类型为循环时，平行扩展是否也具有相同Hopf-Galois结构的问题，采用群论方法并结合Greither-Pareigis和Byott的工作予以解决。
Given an elliptic curve $E$ over a local field $K$ with residue characteristic $3$, we investigate the action of the absolute Galois group of $K$ in the case of potentially good reduction. In particular the only not completely known case is that of the $\ell$-adic Galois representation attached to an elliptic curve such that the image of inertia is non-cyclic, and isomorphic to $C_3 \rtimes C_4$. In this work we describe such a representation explicitly.
We study toric varieties over a field k that split in a Galois extension K/k using Galois cohomology with coefficients in the toric automorphism group. Part of this Galois cohomology fits into an exact sequence induced by the presentation of the class group of the toric variety. This perspective helps to compute the Galois cohomology, particularly for cyclic Galois groups. We use Galois cohomology to classify k-forms of projective spaces when K/k is cyclic, and we also study k-forms of surfaces.
Suppose K is a field of characteristic 0, $K_a$ is its algebraic closure, p is an odd prime. Suppose, $f(x) \in K[x]$ is a polynomial of degree $n \ge 5$ without multiple roots. Let us consider a curve $C: y^p=f(x)$ and its jacobian J(C). It is known that the ring End(J(C)) of all $K_a$-endomorphisms of J(C) contains the ring $Z[ζ_p]$ of integers in the pth cyclotomic field (generated by obvious automorphisms of C). We prove that $End(J(C))=Z[ζ_p]$ if the Galois group of f over K is either the symmetric group $S_n$ or the alternating group $A_n$.
$\DeclareMathOperator{\Aut}{Aut}$Let $p, q$ be distinct primes, with $p > 2$. We classify the Hopf-Galois structures on Galois extensions of degree $p^{2} q$, such that the Sylow $p$-subgroups of the Galois group are cyclic. This we do, according to Greither and Pareigis, and Byott, by classifying the regular subgroups of the holomorphs of the groups $(G, \cdot)$ of order $p^{2} q$, in the case when the Sylow $p$-subgroups of $G$ are cyclic. This is equivalent to classifying the skew braces $(G, \cdot, \circ)$. Furthermore, we prove that if $G$ and $Γ$ are groups of order $p^{2} q$ with non-is…
The interrelation between the cyclic structure of an ideal, i.e., a cyclic code over Galois field $GF(q)$, $q>2$, and its classes of proportional elements is considered. This relation is used in order to define the code's weight structure. The equidistance conditions of irreducible nonprimitive codes over GF(q) are given. Besides that, the minimum distance for some class of nonprimitive cyclic codes is found.
Let $E/F$ be a cyclic Galois extension of degree $p^l$ with Galois group $G$. It is shown that the Galois module structure of both sides of the Kummer pairing (for Kummer extensions of $E$) are the same. In other words, we show that the Kummer duality holds in the level of finitely generated $G$-modules.
Given a quartic Galois extension $L/\mathbb{Q}$ of number fields and a Hopf-Galois structure $H$ on $L/\mathbb{Q}$, we study the freeness of the ring of integers $\mathcal{O}_L$ as module over the associated order $\mathfrak{A}_H$ in $H$. For the classical Galois structure $H_c$, we know by Leopoldt's theorem that $\mathcal{O}_L$ is $\mathfrak{A}_{H_c}$-free. If $L/\mathbb{Q}$ is cyclic, it admits a unique non-classical Hopf-Galois structure, whereas if it is biquadratic, it admits three such Hopf-Galois structures. In both cases, we obtain that freeness depends on the solvability in $\mathbb{…
We investigate the first two Galois cohomology groups of $p$-extensions over a base field which does not necessarily contain a primitive $p$th root of unity. We use twisted coefficients in a systematic way. We describe field extensions which are classified by certain residue classes modulo $p^n$th powers of a related field, and we obtain transparent proofs and slight generalizations of some classical results of Albert. The potential application to the cyclicity question for division algebras of degree $p$ is outlined.
Let $\mathcal{X}$ be a Riemann surface of genus $g>0$ defined over a number field $K$ which is a degree $d$-covering of $\mathbb{P}^1_K$. In this paper we show the existence of infinitely many linearly disjoint degree $d$-extensions $L/K$ over which the Jacobian of $\mathcal{X}$ gains rank. In the case where 0, 1 and $\infty$ are the only branch points, and there is an automorphism $σ$ of $\mathcal{X}$ which cyclically permutes these branch points, we obtain the same result for the Jacobian of $\mathcal{X}/σ$. In particular if $\mathcal{X}$ is the Klein quartic, then the construction provides …
We propose a conjectural characterization of when the dynamical Galois group associated to a polynomial is abelian, and we prove our conjecture in several cases, including the stable quadratic case over ${\mathbb Q}$. In the postcritically infinite case, the proof uses algebraic techniques, including a result concerning ramification in towers of cyclic $p$-extensions. In the postcritically finite case, the proof uses the theory of heights together with results of Amoroso-Zannier and Amoroso-Dvornicich, as well as properties of the Arakelov-Zhang pairing.
Let $G$ be a wreath product of the form $C_2 \wr H$, where $C_2$ is the cyclic group of order 2. Under mild conditions for $H$ we determine the asymptotic behavior of the counting functions for number fields $K/k$ with Galois group $G$ and bounded discriminant. Those counting functions grow linearly with the norm of the discriminant and this result coincides with a conjecture of Malle. Up to a constant factor these groups have the same asymptotic behavior as the conjectured one for symmetric groups.
- Let p be a prime number and K an algebraic number field. What is the arithmetic structure of Galois extensions L/K having p-adic analytic Galois group $Γ$ = Gal(L/K)? The celebrated Tame Fontaine-Mazur conjecture predicts that such extensions are either deeply ramified (at some prime dividing p) or ramified at an infinite number of primes. In this work, we take up a study (initiated by Boston) of this type of question under the assumption that L is Galois over some subfield k of K such that [K : k] is a prime = p. Letting $σ$ be a generator of Gal(K/k), we study the constraints posed on the …
We define a variant of normal basis, called a {\em Galois scaffolding}, that allows for an easy determination of valuation, and has implications for Galois module structure. We identify fully ramified, elementary abelian extensions of local function fields of characteristic $p$, called {\em one-dimensional}, that, in a particular sense, are as simple as cyclic degree $p$ extensions, and prove the statement in the title above.
Cyclic Cubic Points on Higher Genus Curves
本文研究高亏格曲线上伽罗瓦群为 $C_3$ 的三次点，证明其源于结构良好的态射，并给出可计算检验，同时推广至低亏格情形。
Let $p\neq2$ be a prime. We show a technique based on local class field theory and on the expansions of certain resultants which allows to recover very easily Lbekkouri's characterization of Eisenstein polynomials generating cyclic wild extensions of degree $p^2$ over $\Q_p$, and to extend it to the case of the base field $K$ being an unramified extension of $\Q_p$. Furthermore, when a polynomial satisfies only some of the stated conditions, we show that the first unsatisfied condition gives information about the Galois group of the normal closure. This permits to give a complete classificatio…
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