Trinomial extensions of \(\mathbb Q\) with ramification conditions.
Since some years, the Galois group of irreducible trinomials in \(\mathbb{Z}[X]\) have been widely studied; see e.g. \textit{S. D. Cohen}, \textit{A. Movahhedi} and \textit{A. Salinier} [Acta Arith. 82, 1--15 (1997; Zbl 0893.11046); J. Algebra 222, 561--573 (1999; Zbl 0939.12002)] or \textit{A. Hermez} and \textit{A. Salinier} [J. Number Theory 90, 113--129 (2001; Zbl 0990.12004)]. The paper under review concerns Galois extensions over \(\mathbb{Q}\), obtained as splitting fields of rational trinomials, with prescribed ramification behavior at finitely many primes. For a positive integer \(n\) and an arbitrary given finite set \(S\) of prime numbers, the authors consider the existence of degree \(n\) separable monic trinomials \(f(X)\in \mathbb{Z}[X]\) satisfying the following additional properties: (i) the discriminant of \(f(X)\) is not divisible by any prime \(p\in S\), (ii) (resp. (iii)) every prime \(p\in S\) is unramified (resp. tamely ramified) in the splitting field of \(f(X)\) over \(\mathbb{Q}\). The main results characterize pairs of coprime positive integers \(k< n\) for which there exists a trinomial \(X^n +aX^k+ b\in \mathbb{Z}[X]\) whose Galois group over \(\mathbb{Q}\) is contained in \(A_n\) and such that property (i) (resp. (ii); resp. (iii)) holds for a given finite set \(S\). Furthermore, this turns out to be equivalent to requiring that the above Galois group is precisely \(A_n\). In addition, only primes which divide \(n\) or \(k(n- k)\) appear in the conditions obtained.
scientific article; zbMATH DE number 2066769