The asymptotics of nilpotent Galois groups

$\ell$-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.

Cohomology groups of Fermat curves via ray class fields of cyclotomic fields

The absolute Galois group of the cyclotomic field $K={\mathbb Q}(ζ_p)$ acts on the étale homology of the Fermat curve $X$ of exponent $p$. We study a Galois cohomology group which is valuable for measuring an obstruction for $K$-rational points on $X$. We analyze a $2$-nilpotent extension of $K$ which contains the information needed for measuring this obstruction. We determine a large subquotient of this Galois cohomology group which arises from Heisenberg extensions of $K$. For $p=3$, we perform a Magma computation with ray class fields, group cohomology, and Galois cohomology, which determin…

On the generalized Brauer-Siegel theorem for asymptotically exact families with solvable Galois closure

In 2002, M. A. Tsfasman and S. G. Vlăduţ formulated the generalized Brauer-Siegel conjecture for asymptotically exact families of number fields. In this article, we establish this conjecture for asymptotically good towers and asymptotically bad families of number fields with solvable normal closure.

Ramification in the Inverse Galois Problem

This paper focuses on a refinement of the inverse Galois problem. We explore what finite groups appear as the Galois group of an extension of the rational numbers in which only a predetermined set of primes may ramify. After presenting new results regarding extensions in which only a single finite prime ramifies, we move on to studying the more complex situation in which multiple primes from a finite set of arbitrary size may ramify. We then continue by examining a conjecture of Harbater that the minimal number of generators of the Galois group of a tame, Galois extension of the rational numbe…

On unramified solvable extensions of small number fields

We investigate unramified extensions of number fields with prescribed solvable Galois group and certain extra conditions. In particular, we are interested in the minimal degree of a number field $K$, Galois over $\mathbb{Q}$, such that $K$ possesses an unramified $G$-extension. We improve the best known bounds for the degree of such number fields $K$ for certain classes of solvable groups, in particular nilpotent groups.

Galois groups of p-extensions of higher local fields

Suppose $\mathcal K$ is $N$-dimensional local field of characteristic $p$, $\mathcal G =\mathop{Gal}(\mathcal K_{sep}/\mathcal K)$, $\mathcal G_{<p}$ is the maximal quotient of $\mathcal G$ of period $p$ and nilpotent class $<p$ and $\mathcal K_{<p}\subset \mathcal K_{sep}$ is such that $\mathop{Gal}(\mathcal K_{<p}/\mathcal K)=\mathcal G_{<p}$. We use nilpotent Artin-Schreier theory to identify $\mathcal G_{<p}$ with the group $G(\mathcal L)$ obtained from a profinite Lie $\mathbb F_p$-algebra $\mathcal L$ via the Campbell-Hausdorff composition law. The canonical $\mathcal P$-topology on $\ma…

Stickelberger 元的整性与自然 Galois 模的零化

Integrality of Stickelberger elements and annihilation of natural Galois modules

本文研究 Stickelberger 元的系数分母及其对类群和 Quillen K-群的零化作用，将 Brumer、Coates–Sinnott 相关进展推广到幂零扩张，并回答 Greenberg–Gross 问题。

Asymptotic lower bound of class numbers along a Galois representation

Let T be a free Z_p-module of finite rank equipped with a continuous Z_p-linear action of the absolute Galois group of a number field K satisfying certain conditions. In this article, by using a Selmer group corresponding to T, we give a lower bound of the additive p-adic valuation of the class number of K_n, which is the Galois extension field of K fixed by the stabilizer of T/p^n T. By applying this result, we prove an asymptotic inequality which describes an explicit lower bound of the class numbers along a tower K(A[p^\infty])/K for a given abelian variety A with certain conditions in term…

Deformation of Outer Representations of Galois Group II

This paper is devoted to deformation theory of "anabelian" representations of the absolute Galois group landing in outer automorphism group of the algebraic fundamental group of a hyperbolic smooth curve defined over a number-field. In the first part of this paper, we obtained universal deformations for Lie-algebra versions of the above representation using the Schlessinger criteria for functors on Artin local rings. In the second part, we use a version of Schlessinger criteria for functors on the Artinian category of nilpotent Lie algebras which is formulated by Pridham, and explore arithmeti…

Upper bounds on polynomials with small Galois group

When monic integral polynomials of degree $n \geq 2$ are ordered by the maximum of the absolute value of their coefficients, the Hilbert irreducibility theorem implies that asymptotically 100% are irreducible and have Galois group isomorphic to $S_n$. In particular, amongst such polynomials whose coefficients are bounded by $B$ in absolute value, asymptotically $(1+o(1))(2B+1)^n$ are irreducible and have Galois group $S_n$. When $G$ is a proper transitive subgroup of $S_n$, however, the asymptotic count of polynomials with Galois group $G$ has been determined only in very few cases. Here, we s…

钻石：同调与群的中央列

Diamonds: Homology and the Central Series of Groups

研究通过群同调条件，证明非同构数域和超曲面的基本群可有同构的universal nilpotent quotients，揭示了伽罗瓦作用与nilpotent表示论的关系。

The Distribution of Number Fields with Wreath Products as Galois Groups

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.

On some l-adic representations of Gal(Qbar/Q) attached to noncongruence subgroups

The l-adic parabolic cohomology groups attached to noncongruence subgroups of SL_2(Z) are finite-dimensional representations of Gal(Qbar/F) for some number field F. We exhibit examples (with F=Q) giving rise to Galois representations whose images are open subgroups of the full group of symplectic similitudes (of arbitrary dimension). The determination of the image of the Galois group relies on Katz's classification theorem for reductive subalgebras of gl(N) containing a principal nilpotent element; the paper also contains a short conceptual proof of this theorem, suggested by I. Gronowski.

扭变互反多项式的伽罗瓦群：一致渐近

Galois groups of twisted reciprocal polynomials: a uniform asymptotic

本文研究扭变互反多项式的伽罗瓦群，证明 van der Waerden--Bhargava 型渐近定理，给出非完全超八面体群计数公式，并确定首项常数对 $b$ 的依赖。

On counting rings of integers as Galois modules

Let $K$ be a number field and $G$ a finite abelian group. We study the asymptotic behaviour of the number of tamely ramified $G$-extensions of $K$ with ring of integers of fixed realisable class as a Galois module.

Zéro-cycles sur les espaces homogènes et problème de Galois inverse

Let X be a smooth compactification of a homogeneous space of a linear algebraic group G over a number field k. We establish the conjecture of Colliot-Thélène, Sansuc, Kato and Saito on the image of the Chow group of zero-cycles of X in the product of the same groups over all the completions of k. When G is semisimple and simply connected and the geometric stabiliser is finite and supersolvable, we show that rational points of X are dense in the Brauer-Manin set. For finite supersolvable groups, in particular for finite nilpotent groups, this yields a new proof of Shafarevich's theorem on the i…

Local-global compatibility for regular algebraic cuspidal automorphic representation when $\ell \neq p$

We prove the compatibility of local and global Langlands correspondences for $GL_n$ up to semisimplification for the Galois representations constructed by Harris-Lan-Taylor-Thorne and Scholze. More precisely, let $r_p(π)$ denote an $n$-dimensional $p$-adic representation of the Galois group of a CM field $F$ attached to a regular algebraic cuspidal automorphic representation $π$ of $GL_n(\mathbb{A}_F)$. We show that the restriction of $r_p(π)$ to the decomposition group of a place $v\nmid p$ of $F$ corresponds up to semisimplification to $rec(π_v)$, the image of $π_v$ under the local Langlands…

计数函数域的二步幂零野分歧扩张

Counting two-step nilpotent wildly ramified extensions of function fields

本文研究特征 p>2 的函数域中幂零类至多 2 的野分歧扩张的渐近分布，基于最后跳跃不变量计数，通过局部-全局原理和 Abrashkin 幂零 Artin-Schreier 理论得到渐近公式。

椭圆曲线在Galois ℓ-扩张中的Selmer稳定性

Selmer stability for elliptic curves in Galois $\ell$-extensions

本文研究了椭圆曲线在Galois ℓ-扩张中的Selmer稳定性，证明了在特定条件下存在无限多个扩张使Selmer群消失，并给出了渐近下界。

Malle 猜想对于 $\mathbb Q$ 上的 Galois 八次数域

Malle's Conjecture for Galois octic fields over $\mathbb Q$

本文计算了按绝对判别式排序时 Galois 群同构于 $D_4$ 的八次数域的渐近数量，验证了 Malle 强猜想，并证明其比例常数满足 Malle--Bhargava 原理。

简单有理代数扩展的渐进行为

Asymptotics of extensions of simple $\mathbb Q$-algebras

本文研究了简单有理代数扩展的渐进行为，通过分析内Galois扩展和外扩展的分布，将字段扩展的渐进行为研究扩展到非交换情况。

Enumerating $D_4$ Quartics and a Galois Group Bias Over Function Fields

We give an asymptotic formula for the number of $D_4$ quartic extensions of a function field with discriminant equal to some bound, essentially reproducing the analogous result over number fields due Cohen, Diaz y Diaz, and Olivier, but with a stronger error term. We also study the relative density of $D_4$ and $S_4$ quartic extensions of a function field and show that with mild conditions, the number of $D_4$ quartic extensions can far exceed the number of $S_4$ quartic extensions

Galois上同调中的渐近普遍Koszul性

Asymptotic Universal Koszulity in Galois Cohomology

本文研究了格雷德-commutative代数在度1生成下的渐近普遍Koszul性，探讨了其结构性质及在Galois上同调中的应用，证明了有限商的上同调环作为过滤余极限的性质，并提出了局部-整体和修补原理。

Idempotents and Nilpotents Modulo n

We study asymptotic properties of periods and transient phases associated with modular power sequences. The latter are simple; the former are vaguely related to the reciprocal sum of square-free integer kernels.

Multipliers of disposition $p$-groups

Let $p$ be a prime number and $c, d$ natural numbers. Up to isomorphism, there is a unique $p$-group $G^c_d$ of least order with rank $d$ and nilpotency class $c$ named disposition group. This group plays an important role in the construction of Galois extensions over number fields with given $p$-group as Galois group. Also, it has a central series with all factors being elementary. Since $G^c_1$ is abelian we consider $d\geq 2$. In this article, first, we determine the order of all its subgroups of lower central series and $n$-th center subgroups of $G^c_d$, $(n\in\mathbb N)$. Then we deduce …

The shapes of Galois quartic fields

We determine the shapes of all degree $4$ number fields that are Galois. These lie in four infinite families depending on the Galois group and the tame versus wild ramification of the field. In the $V_4$ case, each family is a two-dimensional space of orthorhombic lattices and we show that the shapes are equidistributed, in a regularized sense, in these spaces as the discriminant goes to infinity (with respect to natural measures). We also show that the shape is a complete invariant in some natural families of $V_4$-quartic fields. For $C_4$-quartic fields, each family is a one-dimensional spa…

Asymptotic behavior of class groups and cyclotomic Iwasawa theory of elliptic curves

In this article, we study a relation between certain quotients of ideal class groups and the cyclotomic Iwasawa module $X_\infty$ of the Pontrjagin dual of the fine Selmer group of an elliptic curve $E$ defined over $\mathbb{Q}$. We consider the Galois extension field $K^E_n$ of $\mathbb{Q}$ generated by coordinates of all $p^n$-torsion points of $E$, and introduce a quotient $A^E_n$ of the $p$-sylow subgroup of the ideal class group of $K^E_n$ cut out by the modulo $p^n$ Galois representation $E[p^n]$. We describe the asymptotic behavior of $A^E_n$ by using the Iwasawa module $X_\infty$. In p…

n-Nilpotent Obstructions to pi_1 Sections of P^1-{0,1,infty} and Massey Products

Let pi be a pro-l completion of a free group, and let G be a profinite group acting continuously on pi. First suppose the action is given by a character. Then the boundary maps delta_n: H^1(G, pi/[pi]_n) -> H^2(G, [pi]_n/[pi]_{n+1}) are Massey products. When the action is more general, we partially compute these boundary maps. Via obstructions of Jordan Ellenberg, this implies that pi_1 sections of P^1_k-{0,1,infty} satisfy the condition that associated nth order Massey products in Galois cohomology vanish. For the pi_1 sections coming from rational points, these conditions imply that < (1-x)^…

Galois groups and Cantor actions

In this paper, we study the actions of profinite groups on Cantor sets which arise from representations of Galois groups of certain fields of rational functions. Such representations are associated to polynomials, and they are called profinite iterated monodromy groups. We are interested in a topological invariant of such actions called the asymptotic discriminant. In particular, we give a complete classification by whether the asymptotic discriminant is stable or wild in the case when the polynomial generating the representation is quadratic. We also study different ways in which a wild asymp…

Average Frobenius distribution for elliptic curves defined over finite Galois extensions of the rationals

Let $K$ be a fixed number field, assumed to be Galois over $\mathbb Q$. Let $r$ and $f$ be fixed integers with $f$ positive. Given an elliptic curve $E$, defined over $K$, we consider the problem of counting the number of degree $f$ prime ideals of $K$ with trace of Frobenius equal to $r$. Except in the case $f=2$, we show that "on average," the number of such prime ideals with norm less than or equal to $x$ satisfies an asymptotic identity that is in accordance with standard heuristics. This work is related to the classical Lang-Trotter conjecture and extends the work of several authors.

Malle's conjecture with multiple invariants

We define invariants $\operatorname{inv}_1,\dots,\operatorname{inv}_m$ of Galois extensions of number fields with a fixed Galois group. Then, we propose a heuristic in the spirit of Malle's conjecture which asymptotically predicts the number of extensions that satisfy $\operatorname{inv}_i\leq X_i$ for all $X_i$. The resulting conjecture is proved for abelian Galois groups. We also describe refined Artin conductors that carry essentially the same information as the invariants $\operatorname{inv}_1,\dots,\operatorname{inv}_m$.

Comparing the $π$-primary submodules of the dual Selmer groups

In this paper, we compare the structure of Selmer groups of certain classes of Galois representations over an admissible $p$-adic Lie extension. Namely, we show that the $π$-primary submodules of the Pontryagin dual of the Selmer groups of two Galois representations have the same elementary representations when the two Galois representations in question are either Tate dual to each other or are congruent to each other. In the first situation, our result gives a partial answer to the question of Greenberg on whether the Pontryagin dual of the Selmer groups of two Galois representations that are…

Brauer-Siegel for Arithmetic Tori and lower bounds for Galois orbits of special points

In \cite{S}, Shyr derived an analogue of Dirichlet's class number formula for arithmetic Tori. We use this formula to derive a Brauer-Siegel formula for Tori, relating the Discriminant of a torus to the product of its regulator and class number. We apply this formula to derive asymptotics and lower bounds for Galois orbits of CM points in the Siegel modular variety $A_{g,1}$. Specifically, we ask that the sizes of these orbits grows like a power of Discriminant of the underlying endomorphism algebra. We prove this unconditionally in the case $g\leq 5$, and for all $g$ under the Generalized Rie…

类域塔的渐近增长模式

Asymptotic growth patterns for class field towers

本文研究 $\mathbb{Z}_p$-塔中数域上有限分歧伽罗瓦群的渐近增长模式，将其视为射线类群的非交换类似物，并在特定完全分裂条件下证明了精确的渐近下界，推广了 Iwasawa 关于类数增长的经典结果。

The density of discriminants of quintic rings and fields

We determine, asymptotically, the number of quintic fields having bounded discriminant. Specifically, we prove that the asymptotic number of quintic fields having absolute discriminant at most X is a constant times X. In contrast with the quartic case, we also show that a density of 100% of quintic fields, when ordered by absolute discriminant, have Galois closure with full Galois group $S_5$. The analogues of these results are also proven for orders in quintic fields. Finally, we give an interpretation of the various constants appearing in these theorems in terms of local masses of quintic ri…

Irreducibility of mod p Galois representations of elliptic curves with multiplicative reduction over number fields

In this note we prove that for every integer $d \geq 1$, there exists an explicit constant $B_d$ such that the following holds. Let $K$ be a number field of degree $d$, let $q > \max\{d-1,5\}$ be any rational prime that is totally inert in $K$ and $E$ any elliptic curve defined over $K$ such that $E$ has potentially multiplicative reduction at the prime $\mathfrak q$ above $q$. Then for every rational prime $p> B_d$, $E$ has an irreducible mod $p$ Galois representation. This result has Diophantine applications within the "modular method". We present one such application in the form of an Asymp…

Numbers which are only orders of abelian or nilpotent groups

Refining a result of Erdos and Mays, we give asymptotic series expansions for the functions $A(x)-C(x)$, the count of $n\leq x$ for which every group of order $n$ is abelian (but not all cyclic), and $N(x)-A(x)$, the count of $n\leq x$ for which every group of order $n$ is nilpotent (but not all abelian).

第一伽罗瓦上同调群的调和分析与统计

Harmonic Analysis and Statistics of the First Galois Cohomology Group

本文利用调和分析中的泊松求和与陶伯定理，为第一伽罗瓦上同调群中带类判别式不变量的元素计数问题建立了欧拉乘积分解，从而推进了广义 Malle 猜想。

理想类群的$\boldsymbol{\text{ell}}$-秩的下界

Lower bounds on the $\ell$-rank of ideal class groups

本研究针对数域扩张$K/F$，基于素分歧给出理想类群$\text{ell}$-秩的新下界，该下界适用更广的分歧素理想情形，且在多种伽罗瓦群条件下成立，还用于证明无限类域塔数域的新密度结果。

幂零覆盖的热核与闭测地线渐近性

Heat Kernel and Closed Geodesic Asymptotics for Nilpotent Coverings

研究幂零覆盖上热核及幂零商中闭测地线渐近性，利用有理弗洛凯 - 布洛赫理论等，通过正罗克兰估计等论证给出逐点高阶热核展开式，还产生幂零切博塔廖夫型现象，计算了相关模型。

Semistable abelian varieties with small division fields

Let $A$ be a semistable abelian variety defined over ${\bf Q}$ with bad reduction only at one prime $p$. Let $L= {\bf Q}(A[\ell])$ be the $\ell$-division field of $A$ for a prime $\ell$ not equal to $p$ and let $F={\bf Q}(μ_\ell)$ be the cyclotomic field generated by the group of $\ell^{th}$-roots of unity. We study the varieties $A$ for which $H={\rm Gal(L/F)}$ is "small" in the sense that $H$ is an $\ell$-group or, more generally, that $H$ is nilpotent. We show that if $\ell=2$ or 3 and $H$ is nilpotent then the reduction of $A$ at $p$ is totally toroidal, so its conductor is $p^{\dim A}$. T…

Improved lower bounds for the number of fields with alternating Galois group

Let $n \geq 6$ be an integer. We prove that the number of number fields with Galois group $A_n$ and absolute discriminant at most $X$ is asymptotically at least $X^{1/8 + O(1/n)}$. For $n \geq 8$ this improves upon the previously best known lower bound of $X^{(1 - \frac{2}{n!})/(4n - 4) - ε}$, due to Pierce, Turnage-Butterbaugh, and Wood.

Constructing and counting number fields

In this paper we give a survey of recent methods for the asymptotic and exact enumeration of number fields with given Galois group of the Galois closure. In particular, the case of fields of degree up to 4 is now almost completely solved, both in theory and in practice. The same methods also allow construction of the corresponding complete tables of number fields with discriminant up to a given bound.

系数缓慢增长的多项式的全伽罗瓦群

Full Galois groups of polynomials with slowly growing coefficients

本文研究整数系数首一多项式在系数盒子 $[-L,L]^n$ 中随机选取时伽罗瓦群为全对称群的概率，证明当 $L$ 随次数 $n$ 缓慢趋于无穷时该性质渐近几乎必然成立，并将方法推广到更一般的独立系数情形。

Extensions of number fields with wild ramification of bounded depth

We consider p-extensions of number fields such that the filtration of the Galois group by higher ramification groups is of prescribed finite length. We extend well-known properties of tame extensions to this more general setting; for instance, we show that these towers, when infinite, are ``asymptotically good'' (an explicit bound for the root discriminant is given). We study the difficult problem of bounding the relation-rank of the Galois groups in question. Results of Gordeev and Wingberg imply that the relation-rank can tend to infinity when the set of ramified primes is fixed but the leng…

具有伽罗瓦群 $GL_2(\mathbb{F}_\ell)$ 的数域数量的下界

Lower bounds for the number of number fields with Galois group $GL_2(\mathbb{F}_\ell)$

本文证明了伽罗瓦群为 $GL_2(\mathbb{F}_\ell)$ 且判别式有界的数域数量具有明确的渐近下界，并对按 Artin 导子排序的相应满射伽罗瓦表示给出了类似下界。

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…

On the Galois Theory of Generalized Laguerre Polynomials and Trimmed Exponential

Inspired by the work of Schur on the Taylor series of the exponential and Laguerre polynomials, we study the Galois theory of trimmed exponentials $f_{n,n+k}=\sum_{i=0}^{k} \frac{x^{i}}{(n+i)!}$ and of the generalized Laguerre polynomials $L^{(n)}_k$ of degree $k$. We show that if $n$ is chosen uniformly from $\{1,\ldots, x\}$, then, asymptotically almost surely, for all $k\leq x^{o(1)}$ the Galois groups of $f_{n,n+k}$ and of $L_{k}^{(n)}$ are the full symmetric group $S_k$.

亏格3超椭圆曲线雅可比矩阵上ℓ-挠点的分解模式与定义域域

Factorization patterns and fields of definition of $\ell$-torsion points on the Jacobians of genus 3 hyperelliptic curves

该研究针对有限域上亏格3超椭圆曲线雅可比矩阵的ℓ-挠点问题，通过分析Frobenius作用的若尔当分解，将ℓ-挠点定义域次数的界从O(ℓ⁶)改进到O(ℓ⁴)，并建立了ℓ-挠子群秩与伽罗瓦轨道的对应关系。

6 阶 torsion 和 D_6 扩张的渐进行为

Asymptotics for $6$-torsion and $D_6$-extensions

研究了二次数域类群 6 阶 torsion 的渐进行为，并证明了 Galois D_6 扩张的 Malle 猜想。

数域伽罗瓦扩张的计数

Enumerating Galois extensions of number fields

本文研究数域伽罗瓦扩张的计数问题，通过给出具有给定伽罗瓦群且判别式有界的扩张数量的新上界，得到了判别式不超过X的伽罗瓦扩张数量的渐近公式，改进了Ellenberg和Venkatesh的已有结果。

Torsion bounds for elliptic curves and Drinfeld modules

We derive asymptotically optimal upper bounds on the number of L-rational torsion points on a given elliptic curve or Drinfeld module defined over a finitely generated field K, as a function of the degree [L:K]. Our main tool is the adelic openness of the image of Galois representations attached to elliptic curves and Drinfeld modules, due to Serre and Pink-Ruetsche, respectively. Our approach is to prove a general result for certain Galois modules, which applies simultaneously to elliptic curves and to Drinfeld modules.

弗洛凯-布洛赫理论对幂零群的扩展及其应用

An extension of the Floquet-Bloch theory to nilpotent groups and its applications

本文通过马尔采夫完备化发展离散无挠幂零群的广义弗洛凯-布洛赫理论，得到分支公式并扩展两个经典渐近问题，推导哈珀算子半经典展开，还提出未来研究方向。

具有给定 Galois 群的数域数量的上界

Upper bounds for the number of number fields with prescribed Galois group

本文研究判别式有界且具有给定传递 Galois 群的 $n$ 次数域计数，在特定多元多项式行列式非零的条件下证明其数量的渐近上界，并举例说明行列式计算。