Galois group of \(x^{p^{n}}+aX+a\). (English) Zbl 1528.11118

Benseba, Boualem; Mokhtari, Soufyane

Rocky Mt. J. Math. 53, No. 1, 11-16 (2023).

In this paper, the authors study the Galois group of \(f(X)=X^{p^n}+aX+a\in\mathbb Z[X]\) – an Eisenstein trinomial with respect to \(p\), where \(p\) is an odd prime number and \(n\geq 3\).
In the main result of the present paper, Theorem 3.3, the authors prove that the Galois group \(G\) of \(f(X)\) over the field \(\mathbb Q\) of rational numbers is either the full symmetric group \(S_{p^n}\), or \(AGL(1, p^n) \leq G \leq AGL(n, p)\), where \(AGL(1, p^n)\) is the affine group of dimension 1 over \(\mathbb F_{p^n}\), and \(AGL(n, p)\) is the affine group of dimension \(n\) over \(\mathbb F_p\). They also show that \(G\) is isomorphic to \(S_{p^n}\), except possibly when \(|p^{np^n-1}+\frac{a}{p}(p^n-1)^{p^n-1}|\) is a square, and for each prime divisor \(\ell\) of \(\frac{a}{p}\), \(p\) divides the \(\ell\)-adic valuation \(v_\ell(a)\) of the integer \(a\). Section 2 is devoted to the detailed analysis of different inertia groups of all primes ramified in the splitting field of \(f(X)\) over \(\mathbb Q\).
This study is a natural generalization of studies of Galois groups of Eisenstein trinomials of the type \(X^p+aX+a\). In [J. Algebra 180, No. 3, 966–975 (1996; Zbl 0863.12003)], for example, A. Movahhedi showed that the Galois group \(G\) of this trinomial is either the group Aff(\(\mathbb F_p\)), or the full symmetric group \(S_p\), and the Galois group \(G\) is \(S_p\) when \(a<0\) or \(\frac{a}{p}\not\equiv 1\pmod p\).

Reviewer: Ivo M. Michailov (Shumen)

MSC:

11R32	Galois theory
12F10	Separable extensions, Galois theory
11S15	Ramification and extension theory
12E10	Special polynomials in general fields

Keywords:

Eisenstein trinomials; ramification; Newton polygons; doubly transitive groups; Galois group

Citations:

Zbl 0863.12003

References:

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