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.

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.

scholarly journals Algebraic Properties of a Family of Generalized Laguerre Polynomials

2009 ◽  
Vol 61 (3) ◽  
pp. 583-603 ◽  
Author(s):  
Farshid Hajir
Keyword(s):  
Galois Group ◽  
Laguerre Polynomials ◽  
Irreducible Polynomial ◽  
Main Tool ◽  
Alternating Group ◽  
Newton Polygons ◽  
Generalized Laguerre Polynomials ◽  
Algebraic Properties

Abstract.We study the algebraic properties of Generalized Laguerre Polynomials for negative integral values of the parameter. For integers r, n ≥ 0, we conjecture that is a ℚ-irreducible polynomial whose Galois group contains the alternating group on n letters. That this is so for r = n was conjectured in the 1950's by Grosswald and proven recently by Filaseta and Trifonov. It follows fromrecent work of Hajir andWong that the conjecture is true when r is large with respect to n ≥ 5. Here we verify it in three situations: (i) when n is large with respect to r, (ii) when r ≤ 8, and (iii) when n ≤ 4. The main tool is the theory of p-adic Newton Polygons.

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 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 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>

Infinite Families of A4-Sextic Polynomials

2014 ◽  
Vol 57 (3) ◽  
pp. 538-545 ◽  
Author(s):  
Joshua Ide ◽  
Lenny Jones
Keyword(s):  
Galois Group ◽  
Alternating Group

AbstractIn this article we develop a test to determine whether a sextic polynomial that is irreducible overℚhas Galois group isomorphic to the alternating groupA4. This test does not involve the computation of resolvents, and we use this test to construct several infinite families of such polynomials.

Galois groups of doubly even octic polynomials

2019 ◽  
Vol 19 (01) ◽  
pp. 2050014
Author(s):  
Anna Altmann ◽  
Chad Awtrey ◽  
Sam Cryan ◽  
Kiley Shannon ◽  
Madeleine Touchette
Keyword(s):  
Conjugacy Class ◽  
Galois Group ◽  
Number Field ◽  
Cyclic Group ◽  
Dihedral Group ◽  
Irreducible Polynomial ◽  
Galois Groups ◽  
General Knowledge ◽  
Split Extension

Let [Formula: see text] be an irreducible polynomial with rational coefficients, [Formula: see text] the number field defined by [Formula: see text], and [Formula: see text] the Galois group of [Formula: see text]. Let [Formula: see text], and let [Formula: see text] be the Galois group of [Formula: see text]. We investigate the extent to which knowledge of the conjugacy class of [Formula: see text] in [Formula: see text] determines the conjugacy class of [Formula: see text] in [Formula: see text]. We show that, in general, knowledge of [Formula: see text] does not automatically determine [Formula: see text], except when [Formula: see text] is isomorphic to [Formula: see text] (the cyclic group of order 4). In this case, we show [Formula: see text] is isomorphic to a non-split extension of [Formula: see text] (the dihedral group of order 8) by [Formula: see text]. We also show that [Formula: see text] is completely determined when [Formula: see text] is isomorphic to [Formula: see text] and [Formula: see text] is a perfect square. In this case, [Formula: see text].

Integral Cayley Graphs over Finite Groups

2020 ◽  
Vol 27 (01) ◽  
pp. 131-136
Author(s):  
Elena V. Konstantinova ◽  
Daria Lytkina
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Cayley Graph ◽  
Cyclic Group ◽  
Finite Groups ◽  
Cayley Graphs ◽  
Alternating Group ◽  
Generating Set ◽  
A Finite Group

We prove that the spectrum of a Cayley graph over a finite group with a normal generating set S containing with every its element s all generators of the cyclic group 〈s〉 is integral. In particular, a Cayley graph of a 2-group generated by a normal set of involutions is integral. We prove that a Cayley graph over the symmetric group of degree n no less than 2 generated by all transpositions is integral. We find the spectrum of a Cayley graph over the alternating group of degree n no less than 4 with a generating set of 3-cycles of the form (k i j) with fixed k, as {−n+1, 1−n+1, 22 −n+1, …, (n−1)2 −n+1}.

scholarly journals Mahler measure on Galois extensions

2018 ◽  
Vol 14 (06) ◽  
pp. 1605-1617 ◽  
Author(s):  
Francesco Amoroso
Keyword(s):  
Symmetric Group ◽  
Galois Group ◽  
Galois Extension ◽  
Mahler Measure ◽  
Algebraic Numbers ◽  
Galois Extensions

We study the Mahler measure of generators of a Galois extension with Galois group the full symmetric group. We prove that two classical constructions of generators give always algebraic numbers of big height. These results answer a question of Smyth and provide some evidence to a conjecture which asserts that the height of such a generator grows to infinity with the degree of the extension.

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