Computing Galois groups of certain families of polynomials. (English) Zbl 1473.11207

Shokri, Khosro Monsef; Shaffaf, Jafar; Taleb, Reza

Acta Arith. 185, No. 4, 357-365 (2018).

The topic is the determination of the Galois group of polynomials of the form \[f(x) = x^{n} + a x^{n-1} + b x^{n-2} + c\] with integer coefficients.
As a preliminary result, the authors derive a formula for the discriminant of these polynomials under the assumption that \(a^2 = 4b\).
The main results are sufficient conditions for the Galois group of these polynomials to be the symmetric group \(S_n\). It is shown that this is the case if the respective polynomial is irreducible and the conditions \(c = \pm 1\), \(a^2 = 4b\) and \(\gcd(n,a)= 1\) are satisfied. Sufficient conditions are also given if \(n\) is an odd prime, where the two cases \(a^2 = 4b\) and \(a^2 \neq 4b\) have to be treated separately. The authors show that the techniques used for polynomials of the stated special form can also be applied to a family of polynomials which are not lacunary.
Infinite families of polynomials for which these conditions are satisfied are given.
The first paragraph contains the inaccurate statement that there is no algorithm to calculate the Galois group of a given polynomial in \({\mathbb Z}[x]\). While impractical, such an algorithm is given on page 189 in [B. L. van der Waerden, Modern algebra. Vol. I, translated from the 2nd revised German edition by Fred Blum, with revisions and additions by the author. New York: Frederick Ungar Publishing Co. (1949; Zbl 0039.00902)].

Reviewer: Andreas Bender (Pavia)

MSC:

11R32	Galois theory
11R29	Class numbers, class groups, discriminants
11S15	Ramification and extension theory

Keywords:

Galois groups; polynomials; discriminants; ramification

Citations:

Zbl 0039.00902

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