On Galois Realizations of the 2-Coverable Symmetric and Alternating Groups
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Publication:5409780
DOI10.1080/00927872.2012.713061zbMATH Open1286.11184arXiv1008.3061OpenAlexW2963468822MaRDI QIDQ5409780
Author name not available (Why is that?)
Publication date: 14 April 2014
Published in: (Search for Journal in Brave)
Abstract: Let f(x) be a monic polynomial in Z[x] with no rational roots but with roots in Q_p for all p, or equivalently, with roots mod n for all n. It is known that f(x) cannot be irreducible but can be a product of two or more irreducible polynomials, and that if f(x) is a product of m>1 irreducible polynomials, then its Galois group must be "m-coverable", i.e. a union of conjugates of m proper subgroups, whose total intersection is trivial. We are thus led to a variant of the inverse Galois problem: given an m-coverable finite group G, find a Galois realization of G over the rationals Q by a polynomial f(x) in Z[x] which is a product of m nonlinear irreducible factors (in Q[x]) such that f(x) has a root in Q_p for all p. The minimal value m=2 is of special interest. It is known that the symmetric group S_n is 2-coverable if and only if 2<n<7, and the alternating group A_n is 2-coverable if and only if 3<n<9. In this paper we solve the above variant of the inverse Galois problem for the 2-coverable symmetric and alternating groups, and exhibit an explicit polynomial for each group, with the help of the software packages MAGMA, PARI and GAP.
Full work available at URL: https://arxiv.org/abs/1008.3061
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