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.

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>

A Class of Trinomials with Galois Group Sn

2012 ◽  
Vol 19 (spec01) ◽  
pp. 905-911 ◽  
Author(s):  
Anuj Bishnoi ◽  
Sudesh K. Khanduja
Keyword(s):  
Positive Integer ◽  
Symmetric Group ◽  
Galois Group ◽  
Rational Numbers ◽  
Alternating Group ◽  
Polynomial Formula

A well known result of Schur states that if n is a positive integer and a0, a1,…,an are arbitrary integers with a0an coprime to n!, then the polynomial [Formula: see text] is irreducible over the field ℚ of rational numbers. In case each ai = 1, it is known that the Galois group of fn(x) over ℚ contains An, the alternating group on n letters. In this paper, we extend this result to a larger class of polynomials fn(x) which leads to the construction of trinomials of degree n for each n with Galois group Sn, the symmetric group on n letters.

Galois groups of Taylor polynomials of some elementary functions

2019 ◽  
Vol 15 (06) ◽  
pp. 1127-1141
Author(s):  
Khosro Monsef Shokri ◽  
Jafar Shaffaf ◽  
Reza Taleb
Keyword(s):  
Symmetric Group ◽  
Galois Groups ◽  
Alternating Group ◽  
Integer Number ◽  
Elementary Functions ◽  
Type Formula ◽  
Number Formula ◽  
Taylor Polynomials ◽  
Index 2

Motivated by Schur’s result on computing the Galois groups of the exponential Taylor polynomials, this paper aims to compute the Galois groups of the Taylor polynomials of the elementary functions [Formula: see text] and [Formula: see text]. We first show that the Galois groups of the [Formula: see text]th Taylor polynomials of [Formula: see text] are as large as possible, namely, [Formula: see text] (full symmetric group) or [Formula: see text] (alternating group), depending on the residue of the integer number [Formula: see text] modulo [Formula: see text]. We then compute the Galois groups of the [Formula: see text]th Taylor polynomials of [Formula: see text] and show that these Galois groups essentially coincide with the Coexter groups of type [Formula: see text] (or an index 2 subgroup of the corresponding Coexter group).

Infinite families of monogenic trinomials and their Galois groups

2018 ◽  
Vol 29 (05) ◽  
pp. 1850039 ◽  
Author(s):  
Lenny Jones ◽  
Tristan Phillips
Keyword(s):  
Positive Integer ◽  
Symmetric Group ◽  
Galois Groups ◽  
Alternating Group

Let [Formula: see text] with [Formula: see text]. Let [Formula: see text] and [Formula: see text] denote, respectively, the symmetric group and alternating group on [Formula: see text] letters. Let [Formula: see text] be an indeterminate, and define [Formula: see text] where [Formula: see text] are certain prescribed forms in [Formula: see text]. For a certain set of these forms, we show unconditionally that there exist infinitely many primes [Formula: see text] such that [Formula: see text] is irreducible over [Formula: see text], [Formula: see text], and the fields [Formula: see text] are distinct and monogenic, where [Formula: see text]. Using a different set of forms, we establish a similar result for all square-free values of [Formula: see text], with [Formula: see text], and any positive integer value of [Formula: see text] for which [Formula: see text] is square-free. Additionally, in this case, we prove that [Formula: see text]. Finally, we show that these results can be extended under the assumption of the [Formula: see text]-conjecture. Our methods make use of recent results of Helfgott and Pasten.

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.

scholarly journals The Galois group of f(xr)

1984 ◽  
Vol 25 (1) ◽  
pp. 75-91 ◽  
Author(s):  
S. D. Cohen ◽  
W. W. Stothers
Keyword(s):  
Symmetric Group ◽  
Galois Group ◽  
Irreducible Polynomial ◽  
Alternating Group

Let f(x) be an irreducible polynomial of degree n with coefficients in a field L and r be an integer prime to the characteristic of L. The object of this paper is to describe the galois group g of f(xr) over L when the galois group G of f(x) itself over L is either the full symmetric group Snor the alternating group An. We shall call f standard if G = Sn or An with |G|>2.

scholarly journals Rational Trinomials with the Alternating Group as Galois Group

2001 ◽  
Vol 90 (1) ◽  
pp. 113-129 ◽  
Author(s):  
Alain Hermez ◽  
Alain Salinier
Keyword(s):  
Galois Group ◽  
Alternating Group

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.

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