×

Tame \(A_{n}\)-extensions of \({\mathbb Q}\). (English) Zbl 1057.12003

The authors prove the following result: For every positive integer \(n\) and every finite set \(S\) of prime numbers, there exist infinitely many disjoint extensions of \({\mathbb Q},\) each one obtained as the splitting field over \({\mathbb Q}\) of a totally real monic polynomial \(f(x)\in Z[x]\) of degree \(n\) such that the discriminant of \(f(x)\) is not divisible by any prime number of \(S\) and such that \(f(x)\) has Galois group \(A_n\) over \({\mathbb Q}.\)

MSC:

12F12 Inverse Galois theory
11R09 Polynomials (irreducibility, etc.)
11R32 Galois theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Birch, B., Noncongruence subgroups, covers and drawings, (Schneps, L., The Grothendieck Theory of Dessins d’Enfants (1994), Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 25-46 · Zbl 0930.11024
[2] Mestre, J.-F., Extensions régulières de \(Q(T)\) de groupe de Galois \(An\), J. Algebra, 131, 483-495 (1990) · Zbl 0714.11074
[3] Morita, Y., A Note on the Hilbert irreducibility theorem, Proc. Japan Acad. Ser. A Math. Sci., 66, 101-104 (1990) · Zbl 0725.12003
[4] Narkiewicz, W., Elementary and Analytic Theory of Algebraic Numbers (1990), Springer, PWN—Polish Scientific Publishers · Zbl 0717.11045
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.