Safarevic's theorem on solvable groups as Galois groups

A non-solvable extension of $\Q$ unramified outside 7

We consider a mod 7 Galois representation attached to a genus 2 Siegel cuspforms of level 1 and weight 28 and using some of its Fourier coefficients and eigenvalues computed by N. Skoruppa and the classification of maximal subgroups of PGSp(4,p) we show that its image is as large as possible. This gives a realization of PGSp(4,7) as a Galois group over $\Q$ and the corresponding number field provides a non-solvable extension of $\Q$ which ramifies only at 7.

作为可解扩张上Hopf-Galois结构的类型出现的非阿贝尔单群

Non-abelian simple groups which occur as the type of a Hopf-Galois structure on a solvable extension

本研究确定了可解扩张上作为Hopf-Galois结构类型的有限非阿贝尔单群，同时给出了乘法群可解的斜brace中作为加法群的这类单群的完整列表。

具有正规阿贝尔子群的可解域类群的无限矩

Infinite Moments of Class Groups for Solvable Fields with a Normal Abelian Subgroup

利用类域论和Minkowski界，在Galois群可解且满足适当条件时，估计受限分歧解的数量，从而证明特定域计数问题的上界及类群的无限矩。

Galois extensions ramified only at one prime

This paper gives some restrictions on finite groups that can occur as Galois groups of extensions over $\Q$ and over $\F_q(t)$ ramified only at one finite prime. Over $\Q$, we strengthen results of Jensen and Yui about dihedral extensions and rule out some non-solvable groups. Over $\F_q(t)$ restrictions are given for symmetric groups and dihedral groups to occur as tamely ramified extension over $\F_q(t)$ ramified only at one prime.

Galois理论

Galois Theory

本研究是爱丁堡大学2021-2023年本科Galois理论课程讲义，以代数基础知识为前提，讲解域扩张与Galois群理论至基本定理，还包含尺规作图、可解多项式相关证明及有限域分类。

数论中绝对伽罗瓦群的Ab-忠实性

Ab-faithfulness of absolute Galois groups in number theory

本文证明了混合特征离散赋值域及正维数高维局部域的绝对伽罗瓦群是ab-忠实的，并给出形式Hilbert~90判据以推导其瘦性。

On Galois extensions with prescribed decomposition groups

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…

$\ell$-torsion bounds for the class group of number fields with an $\ell$-group as Galois group

We describe the relations among the $\ell$-torsion conjecture, a conjecture of Malle giving an upper bound for the number of extensions, and the discriminant multiplicity conjecture. We prove that the latter two conjectures are equivalent in some sense. Altogether, the three conjectures are equivalent for the class of solvable groups. We then prove the $\ell$-torsion conjecture for $\ell$-groups and the other two conjectures for nilpotent groups.

On finite embedding problems with abelian kernels

Given a Hilbertian field $k$ and a finite set $\mathcal{S}$ of Krull valuations of $k$, we show that every finite split embedding problem $G \rightarrow {\rm{Gal}}(L/k)$ over $k$ with abelian kernel has a solu\-tion ${\rm{Gal}}(F/k) \rightarrow G$ such that every $v \in \mathcal{S}$ is totally split in $F/L$. Two applications are then given. Firstly, we solve a non-constant variant of the Beckmann--Black problem for solvable groups: given a field $k$ and a non-trivial finite solvable group $G$, every Galois field extension $F/k$ of group $G$ is shown to occur as the specialization at some $t_0…

The unramified inverse Galois problem and cohomology rings of totally imaginary number fields

We employ methods from homotopy theory to define new obstructions to solutions of embedding problems. By using these novel obstructions we study embedding problems with non-solvable kernel. We apply these obstructions to study the unramified inverse Galois problem. That is, we show that our methods can be used to determine that certain groups cannot be realized as the Galois groups of unramified extensions of certain number fields. To demonstrate the power of our methods, we give an infinite family of totally imaginary quadratic number fields such that $\text{Aut}(\text{PSL}(2,q^2))$ for $q$ a…

The $m$-step solvable anabelian geometry of number fields

Given a number field $K$ and an integer $m\geq 0$, let $K_m$ denote the maximal $m$-step solvable Galois extension of $K$ and write $G_K^m$ for the maximal $m$-step solvable Galois group Gal$(K_m/K)$ of $K$. In this paper, we prove that the isomorphy type of $K$ is determined by the isomorphy type of $G_K^3$. Further, we prove that $K_m/K$ is determined functorially by $G_K^{m+3}$ (resp. $G_K^{m+4}$) for $m\geq 2$ (resp. $m \leq 1$). This is a substantial sharpening of a famous theorem of Neukirch and Uchida. A key step in our proof is the establishment of the so-called local theory, which in …

Upper bound on the number of ramified primes for odd order solvable groups

Let $G$ be a finite group and let $ram^{t}(G)$ denote the minimal positive integer $n$ such that $G$ can be realized as the Galois group of a tamely ramified extension of $\mathbb{Q}$ ramified only at $n$ finite primes. Let $d(G)$ denote the minimal non negative integer for which there exists a subset $X$ of $G$ with $d(G)$ elements such that the normal subgroup of $G$ generated by $X$ is all of $G$. It is known that $d(G)\leq ram^{t}(G)$. However, it is unknown whether or not every finite group $G$ can be realized as a Galois group of a tamely ramified extension of $\mathbb{Q}$ with exactly $…

Formal group exponentials and Galois modules in Lubin-Tate extensions

Explicit descriptions of local integral Galois module generators in certain extensions of $p$-adic fields due to Pickett have recently been used to make progress with open questions on integral Galois module structure in wildly ramified extensions of number fields. In parallel, Pulita has generalised the theory of Dwork's power series to a set of power series with coefficients in Lubin-Tate extensions of $\Q_p$ to establish a structure theorem for rank one solvable p-adic differential equations. In this paper we first generalise Pulita's power series using the theories of formal group exponent…

可解下降与可解群的Grünwald问题

Solvable Descent and the Grunwald Problem for Solvable Groups

本文通过Harpaz和Wittenberg的方法证明了可解下降定理，解决了可解群的Grünwald问题，并提供了Shafarevich逆伽罗瓦问题的替代证明。

On Galois Groups of Prime Degree Polynomials with Complex Roots

Let $f$ be an irreducible polynomial of prime degree $p\geq 5$ over $\QQ$, with precisely $k$ pairs of complex roots. Using a result of Jens Höchsmann (1999), we show that if $p\geq 4k+1$ then $\Gal(f/\QQ)$ is isomorphic to $A_{p}$ or $S_{p}$. This improves the algorithm for computing the Galois group of an irreducible polynomial of prime degree, introduced by A. Bialostocki and T.Shaska. If such a polynomial $f$ is solvable by radicals then its Galois group is a Frobenius group of degree p. Conversely, any Frobenius group of degree p and of even order, can be realized as the Galois group of a…

The Sylow subgroups of the absolute Galois group Gal(Q)

We describe the Sylow subgroups of Gal(Q) for an odd prime p, by observing and studying their decomposition as a semidirect product of Z_p acting on F, where F is a free pro-p group, and Z_p are the p-adic integers. We determine the finite Z_p-quotients of F and more generally show that every split embedding problem of Z_p-groups for F is solvable. Moreover, we analyze the Z_p-action on generators of F.

Tamely ramified subfields of division algebras

For any number field K, it is unknown which finite groups appear as Galois groups of extensions L/K such that L is a maximal subfield of a division algebra with center K (a K-division algebra). For K=Q, the answer is described by the long standing Q-admissibility conjecture. We extend a theorem of Neukirch on embedding problems with local constraints in order to determine for every number field K, what finite solvable groups G appear as Galois groups of tame maximal subfields of K-division algebras, generalizing Liedahl's theorem for metacyclic G and Sonn's solution of the Q-admissibility conj…

Galois-theoretic features for 1-smooth pro-$p$ groups

Let $p$ be a prime. A pro-$p$ group $G$ is said to be 1-smooth if it can be endowed with a continuous representation $θ\colon G\to\mathrm{GL}_1(\mathbb{Z}_p)$ such that every open subgroup $H$ of $G$, together with the restriction $θ\vert_H$, satisfies a formal version of Hilbert 90. We prove that every 1-smooth pro-$p$ group contains a unique maximal closed abelian normal subgroup, in analogy with a result by Engler and Koenigsmann on maximal pro-$p$ Galois groups of fields, and that if a 1-smooth pro-$p$ group is solvable, then it is locally uniformly powerful, in analogy with a result by Wa…

Chebyshev polynomials and Galois groups of De Moivre polynomials

Let $n\ge 3$ be an odd natural number. In 1738, Abraham de Moivre introduced a family of polynomials of degree $n$ with rational coefficients, all of which are solvable. So far, the Galois groups of these polynomials have been investigated only for prime numbers $n$ and under special assumptions. We describe the Galois groups for arbitrary odd $n\ge 3$ in the irreducible case, up to few exceptions. In addition, we express all zeros of such a polynomial as rational functions of three zeros, two of which are connected in a certain sense. These results are based on the reduction of the radical $$…

Solvable points on genus one curves over local fields

Let $F$ be a field complete with respect to a discrete valuation whose residue field is perfect of characteristic $p>0$. We prove that every smooth, projective, geometrically irreducible curve of genus one defined over $F$ with a non-zero divisor of degree a power of $p$ has a solvable point over $F$. We also show that there is a field $F$ complete with respect to a discrete valuation whose residue field is perfect and there is a finite Galois extension $K|F$ such that there is no solvable extension $L|F$ such that the extension $KL|K$ is unramified, where $KL$ is the composite of $K$ and $L$.…

Polynomials with roots in ${\Bbb Q}_p$ for all $p$

Let $f(x)$ be a monic polynomial in $\dZ[x]$ with no rational roots but with roots in $\dQ_p$ for all $p$, or equivalently, with roots mod $n$ for all $n$. It is known that $f(x)$ cannot be irreducible but can be a product of two or more irreducible polynomials, and that if $f(x)$ is a product of $m>1$ irreducible polynomials, then its Galois group must be a union of conjugates of $m$ proper subgroups. We prove that for any $m>1$, every finite solvable group which is a union of conjugates of $m$ proper subgroups (where all these conjugates have trivial intersection) occurs as the Galois group …

Detecting Fast solvability of equations via small powerful Galois groups

Fix an odd prime $p$, and let $F$ be a field containing a primitive $p$th root of unity. It is known that a $p$-rigid field $F$ is characterized by the property that the Galois group $G_F(p)$ of the maximal $p$-extension $F(p)/F$ is a solvable group. We give a new characterization of $p$-rigidity which says that a field $F$ is $p$-rigid precisely when two fundamental canonical quotients of the absolute Galois groups coincide. This condition is further related to analytic $p$-adic groups and to some Galois modules. When $F$ is $p$-rigid, we also show that it is possible to solve for the roots o…

Statistics of the First Galois Cohomology Group: A Refinement of Malle's Conjecture

Malle proposed a conjecture for counting the number of $G$-extensions $L/K$ with discriminant bounded above by $X$, denoted $N(K,G;X)$, where $G$ is a fixed transitive subgroup $G\subset S_n$ and $X$ tends towards infinity. We introduce a refinement of Malle's conjecture, if $G$ is a group with a nontrivial Galois action then we consider the set of crossed homomorphisms in $Z^1(K,G)$ (or equivalently $1$-coclasses in $H^1(K,G)$) with bounded discriminant. This has a natural interpretation given by counting $G$-extensions $F/L$ for some fixed $L$ and prescribed extension class $F/L/K$. If $T$ i…

A Theorem on GL(n) a la Tchebotarev

Let $K/F$ be a finite Galois extension of number fields. It is well known that the Tchebotarev density theorem implies that an irreducible, finitely ramified $p$-adic representation $ρ$ of the absolute Galois group of $K$ is determined (up to equivalence) by the characteristic polynomials of Frobenius elements Fr$_v$ at any set of primes $v$ of $K$ of degree $1$ over $F$. Here we prove an analogue for GL$(n)$, namely that a cuspidal automorphic representation $π$ of GL$(n, {\mathbb A}_K)$ is determined up by the knowledge of its local components at the primes of degree one over $F$. We prove i…

Galois theory for general systems of polynomial equations

We prove that the monodromy group of a reduced irreducible square system of general polynomial equations equals the symmetric group. This is a natural first step towards the Galois theory of general systems of polynomial equations, because arbitrary systems split into reduced irreducible ones upon monomial changes of variables. In particular, our result proves the multivariate version of the Abel--Ruffini theorem: the classification of general systems of equations solvable by radicals reduces to the classification of lattice polytopes of mixed volume 4 (which we prove to be finite in every dim…

混合特征局部域的$m$-步可解非阿贝尔几何

The $m$-step solvable anabelian geometry of mixed-characteristic local fields

本文针对混合特征局部域$K$，证明其同构类由极大2-步可解商群$G_K^2$的滤过完备群同构类决定，并给出$K^m/K$由$G_K^{m+2}$或$G_K^{m+3}$函子决定的结论。

数域上最大Kummer扩张中概形的上同调

Cohomology of varieties over the maximal Kummer extension of a number field

本文研究了数域上最大Kummer扩张中概形的上同调性质，证明了其有限不变类及阿贝尔簇 torsion 的有限性，同时指出非阿贝尔可解扩张的torsion有限性不受伽罗瓦群控制。

关于素数次数不可解方程

On unsolvable equations of prime degree

本文基于Kronecker定理，通过修正Weber证明中的错误，给出一个简洁的证明，用于构造整数系数奇素数次数不可解方程的例子。

On the Malle conjecture and the Grunwald problem

We contribute to the Malle conjecture on the number N (K, G, y) of finite Galois extensions E of some number field K of finite group G and of discriminant of norm |N K/Q (d E)| $\le$ y. We prove the lower bound part of the conjecture for every group G and every number field K containing a certain number field K 0 depending on G : N (K, G, y) $\ge$ y $α$(G) for y 1 and some specific exponent $α$(G) depending on G. To achieve this goal, we start from a regular Galois extension F/K(T) that we specialize. We prove a strong version of the Hilbert Irreducibility Theorem which counts the number of sp…

数域上的可容许群

Admissible groups over number fields

本文研究数域上哪些有限群可作为可除代数极大子域的伽罗瓦群，在驯顺分歧情形给出使所有可解 Sylow-亚循环群可容许的数域完全分类，并在野分歧情形给出若干类数域的特征刻画与部分结果。