scholarly journals On the Galois group of three classes of trinomials

2021 ◽  
Vol 7 (1) ◽  
pp. 212-224
Author(s):  
Lingfeng Ao ◽  
◽  
Shuanglin Fei ◽  
Shaofang Hong
Keyword(s):  
Symmetric Group ◽  
Galois Group ◽  
Prime Number ◽  
Galois Groups

<abstract><p>Let $ n\ge 8 $ be an integer and let $ p $ be a prime number satisfying $ \frac{n}{2} &lt; p &lt; n-2 $. In this paper, we prove that the Galois groups of the trinomials</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ T_{n, p, k}(x): = x^n+n^kp^{(n-1-p)k}x^p+n^kp^{nk}, $\end{document} </tex-math></disp-formula></p> <p><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ S_{n, p}(x): = x^n+p^{n(n-1-p)}n^px^p+n^pp^{n^2} $\end{document} </tex-math></disp-formula></p> <p>and</p> <p><disp-formula> <label/> <tex-math id="FE3"> \begin{document}$ E_{n, p}(x): = x^n+pnx^{n-p}+pn^2 $\end{document} </tex-math></disp-formula></p> <p>are the full symmetric group $ S_n $ under several conditions. This extends the Cohen-Movahhedi-Salinier theorem on the irreducible trinomials $ f(x) = x^n+ax^s+b $ with integral coefficients.</p></abstract>

scholarly journals One-Relator Maximal Pro-p Galois Groups and the Koszulity Conjectures

2020 ◽  
Author(s):  
Claudio Quadrelli
Keyword(s):  
Galois Group ◽  
Prime Number ◽  
Group Algebra ◽  
Galois Groups ◽  
Cohomology Algebra ◽  
Augmentation Ideal ◽  
Graded Algebra ◽  
Absolute Galois Group ◽  
Quadratic Algebras ◽  
Elementary Type

Abstract Let p be a prime number and let ${\mathbb{K}}$ be a field containing a root of 1 of order p. If the absolute Galois group $G_{\mathbb{K}}$ satisfies $\dim\, H^1(G_{\mathbb{K}},\mathbb{F}_p)\lt\infty$ and $\dim\, H^{\,2}(G_{\mathbb{K}},\mathbb{F}_p)=1$, we show that L. Positselski’s and T. Weigel’s Koszulity conjectures are true for ${\mathbb{K}}$. Also, under the above hypothesis, we show that the $\mathbb{F}_p$-cohomology algebra of $G_{\mathbb{K}}$ is the quadratic dual of the graded algebra ${\rm gr}_\bullet\mathbb{F}_p[G_{\mathbb{K}}]$, induced by the powers of the augmentation ideal of the group algebra $\mathbb{F}_p[G_{\mathbb{K}}]$, and these two algebras decompose as products of elementary quadratic algebras. Finally, we propose a refinement of the Koszulity conjectures, analogous to I. Efrat’s elementary type conjecture.

Totally Real Subfields of p-Adic Fields having the Symmetric Group as Galois Group

1971 ◽  
Vol 14 (3) ◽  
pp. 441-442 ◽  
Author(s):  
Howard Kleiman
Keyword(s):  
Symmetric Group ◽  
Galois Group ◽  
Elementary Proof ◽  
Symmetric Groups ◽  
Rational Numbers ◽  
Galois Groups ◽  
Arbitrary Field ◽  
Totally Real

In this paper, an elementary proof is given of the following proposition:Theorem 1. If Qp is an arbitrary field of p-adic numbers, then it contains normal subfields Ln(2 ≤ n ≤ p) which have symmetric groups Sn as their respective Galois groups over Q, the field of rational numbers. Furthermore, each Ln may be chosen to be totally real.

WILD RAMIFICATION IN TRINOMIAL EXTENSIONS AND GALOIS GROUPS

2020 ◽  
Vol 63 (1) ◽  
pp. 106-120
Author(s):  
BOUALEM BENSEBAA ◽  
ABBAS MOVAHHEDI ◽  
ALAIN SALINIER
Keyword(s):  
Symmetric Group ◽  
Galois Group ◽  
Galois Groups ◽  
Alternating Group ◽  
Prime Degree ◽  
Wild Ramification ◽  
Wide Range

AbstractIt is proven that, for a wide range of integers s (2 < s < p − 2), the existence of a single wildly ramified odd prime l ≠ p leads to either the alternating group or the full symmetric group as Galois group of any irreducible trinomial Xp + aXs + b of prime degree p.

Pointwise bound for ℓ-torsion in class groups: Elementary abelian extensions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Jiuya Wang
Keyword(s):  
Galois Group ◽  
Prime Number ◽  
Finite Groups ◽  
Upper Bound ◽  
Abelian Groups ◽  
Class Groups ◽  
Abelian Extensions ◽  
Pointwise Bound ◽  
First Case

AbstractElementary abelian groups are finite groups in the form of {A=(\mathbb{Z}/p\mathbb{Z})^{r}} for a prime number p. For every integer {\ell>1} and {r>1}, we prove a non-trivial upper bound on the {\ell}-torsion in class groups of every A-extension. Our results are pointwise and unconditional. This establishes the first case where for some Galois group G, the {\ell}-torsion in class groups are bounded non-trivially for every G-extension and every integer {\ell>1}. When r is large enough, the unconditional pointwise bound we obtain also breaks the previously best known bound shown by Ellenberg and Venkatesh under GRH.

scholarly journals Calculating Galois groups of third-order linear differential equations with parameters

2018 ◽  
Vol 20 (04) ◽  
pp. 1750038
Author(s):  
Andrei Minchenko ◽  
Alexey Ovchinnikov
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Galois Group ◽  
Galois Groups ◽  
Linear Differential Equations ◽  
Unipotent Radical ◽  
Third Order ◽  
Differential Galois Group ◽  
Bessel Differential Equation ◽  
Linear Differential

Motivated by developing algorithms that decide hypertranscendence of solutions of extensions of the Bessel differential equation, algorithms computing the unipotent radical of a parameterized differential Galois group have been recently developed. Extensions of Bessel’s equation, such as the Lommel equation, can be viewed as homogeneous parameterized linear differential equations of the third order. In this paper, we give the first known algorithm that calculates the differential Galois group of a third-order parameterized linear differential equation.

Endomorphisms of Two Dimensional Jacobians and Related Finite Algebras

2012 ◽  
Vol 55 (1) ◽  
pp. 38-47
Author(s):  
William Butske
Keyword(s):  
Galois Group ◽  
Galois Groups ◽  
Two Dimensional ◽  
Finite Algebras ◽  
Genus 2

AbstractZarhin proves that if C is the curve y2 = f (x) where Galℚ(f(x)) = Sn or An, then . In seeking to examine his result in the genus g = 2 case supposing other Galois groups, we calculate for a genus 2 curve where f (x) is irreducible. In particular, we show that unless the Galois group is S5 or A5, the Galois group does not determine .

scholarly journals On Galois groups of class two extensions over the rational number field

1979 ◽  
Vol 75 ◽  
pp. 121-131 ◽  
Author(s):  
Susumu Shirai
Keyword(s):  
Nilpotent Group ◽  
Galois Group ◽  
Number Field ◽  
Direct Product ◽  
Rational Number ◽  
Nilpotency Class ◽  
Abelian Extension ◽  
Galois Groups ◽  
Classification Theory ◽  
Ray Class Field

Let Q be the rational number field, K/Q be a maximal Abelian extension whose degree is some power of a prime l, and let f(K) be the conductor of K/Q; if l = 2, let K be complex, and if in addition f(K) ≡ 0 (mod 2), let f(K) ≡ 0 (mod 16). Denote by (K) the Geschlechtermodul of K over Q and by K̂ the maximal central l-extension of K/Q contained in the ray class field mod (K) of K. A. Fröhlich [1, Theorem 4] completely determined the Galois group of K̂ over Q in purely rational terms. The proof is based on [1, Theorem 3], though he did not write the proof in the case f(K) ≡ 0 (mod 16). Moreover he gave a classification theory of all class two extensions over Q whose degree is a power of l. Hence we know the set of fields of nilpotency class two over Q, because a finite nilpotent group is a direct product of all its Sylow subgroups. But the theory becomes cumbersome, and it is desirable to reconstruct a more elementary one.

scholarly journals Galois Group of the Maximal Abelian Extension over an Algebraic Number Field

1957 ◽  
Vol 12 ◽  
pp. 177-189 ◽  
Author(s):  
Tomio Kubota
Keyword(s):  
Field Theory ◽  
Galois Group ◽  
Prime Number ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Abelian Extension ◽  
Finite Degree ◽  
Class Field ◽  
Continuous Character

The aim of the present work is to determine the Galois group of the maximal abelian extension ΩA over an algebraic number field Ω of finite degree, which we fix once for all.Let Z be a continuous character of the Galois group of ΩA/Ω. Then, by class field theory, the character Z is also regarded as a character of the idele group of Ω. We call such Z character of Ω. For our purpose, it suffices to determine the group Xl of the characters of Ω whose orders are powers of a prime number l.

scholarly journals Computing the Galois group of a polynomial over a p-adic field

2020 ◽  
Vol 16 (08) ◽  
pp. 1767-1801 ◽  
Author(s):  
Christopher Doris
Keyword(s):  
Galois Group ◽  
Galois Groups ◽  
Naive Algorithm

We present a family of algorithms for computing the Galois group of a polynomial defined over a p-adic field. Apart from the “naive” algorithm, these are the first general algorithms for this task. As an application, we compute the Galois groups of all totally ramified extensions of [Formula: see text] of degrees 18, 20 and 22, tables of which are available online.

scholarly journals GALOIS GROUPS AND AN OBSTRUCTION TO PRINCIPAL GRAPHS OF SUBFACTORS

2007 ◽  
Vol 18 (02) ◽  
pp. 191-202 ◽  
Author(s):  
MARTA ASAEDA
Keyword(s):  
Recent Result ◽  
Galois Group ◽  
Numerical Computation ◽  
Galois Groups ◽  
Jones Index ◽  
Principal Graph ◽  
New Type ◽  
Index Value

The Galois group of the minimal polymonal of a Jones index value gives a new type of obstruction to a principal graph, thanks to a recent result of Etingof, Nikshych, and Ostrik. We show that the sequence of the graphs given by Haagerup as candidates of principal graphs of subfactors, are not realized as principal graphs for 7 < n ≤ 27 using GAP program. We further utilize Mathematica to extend the statement to 27 < n ≤ 55. We conjecture that none of the graphs are principal graphs for all n > 7, and give an evidence using Mathematica for smaller graphs among them for n > 55. The problem for the case n = 7 remains open, however, it is highly likely that it would be realized as a principal graph, thanks to numerical computation by Ikeda.

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