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Arithmetic of octahedral sextics. (English) Zbl 1300.11108

Let \(K/k\) be a Galois extension of degree \(24\) with group \(S_4\). Assuming char \(k\neq2\) the author shows (Theorems 1 and 2) how related are the generators of quartic and sextic subfields of \(K\). The next results deal with the case when \(k\) is the field of rationals. Theorem 4 determines the tame parts of discriminants of subfields of \(K\), and Theorem 6 establishes the existence of infinitely many \(K\) having a quartic subfield with discriminant of the form \(-A^2\) (\(A\in \mathbb Z\)). In Theorem 3 it is proved that there are exactly six sextic extensions of the rationals with group \(S_4\), where only the prime \(229\) is ramified.
The author considers also quartic subfields of \(K\) with prime power discriminants, and gives examples which cannot be obtained with the method used by D. Doud [J. Number Theory 75, No. 2, 185–197 (1999; Zbl 0928.11049)] to construct such fields.
Finally, two interesting conjectures are proposed, one of them concerning the number of \(S_4\)-extensions \(K/Q\) with prime-power \(|d(K)|\) containing a fixed non-abelian cubic subfield \(L\) with prime-power \(|d(L)|\).

MSC:

11R16 Cubic and quartic extensions
11R09 Polynomials (irreducibility, etc.)
11R21 Other number fields
11R29 Class numbers, class groups, discriminants
11R32 Galois theory

Citations:

Zbl 0928.11049

Software:

PARI/GP; nftables
PDFBibTeX XMLCite
Full Text: DOI

References:

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