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Double transitivity of Galois groups of trinomials. (English) Zbl 0893.11046

This paper studies the double transitivity of Galois groups over \(\mathbb{Q}\) for irreducible trinomials as permutation groups on their zeros. Let \(f(x)= X^n+ aX^s+b\) be an irreducible trinomial with integral coefficients, \(1\leq s\leq n-1\), \(ab\neq 0\). The main criterion obtained is the following: If \((n,as)= (a(n-s),b) =1\) and there exists a prime divisor \(p\) of \(b\) such that \((s,v_p(b)) =1\), then the Galois group \(G(f)\) of \(f(X)\) over \(\mathbb{Q}\) is doubly transitive. Since a doubly transitive group of degree \(n\) which contains a transposition is the full symmetric group \(S_n\), the authors obtain explicit conditions under which the Galois group of an irreducible trinomial is \(S_n\), improving earlier results of H. Osada [J. Number Theory 25, 230-238 (1987; Zbl 0608.12010)].
The authors study the inertia groups of primes \(p\) in the splitting field of these trinomials \(f(X)\) related to their factorization over \(\mathbb{Q}_p [X]\); they then show that the Galois groups \(G(f)\) of the trinomials considered contain a subgroup acting transitively on a suitable set of roots of \(f(X)\) and that the Galois groups \(G(f)\) are primitive permutation groups. This gives the double transitivity of \(G(f)\) by Jordan’s theorem. Using similar methods, A. Movahhedi and A. Salinier have obtained primitivity criteria for irreducible trinomials [cf. J. Lond. Math. Soc., II. Ser. 53, 433-490 (1996; Zbl 0862.11063)].
Reviewer: N.Vila (Barcelona)

MSC:

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