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Division algebras with \(\text{PSL} (2,q)\)-Galois maximal subfields. (English) Zbl 0977.12004

Let \(G\) be a finite group and let \(k\) be a field. Then \(G\) is called \(k\)-admissible if there exists a \(G\)-Galois extension \(L/k\) such that \(L\) is a maximal subfield of a \(k\)-division algebra. The main result of the paper under review states that \({\text{PSL}}(2,7)\) is \(k\)-admissible for any number field \(k\) for which \(\sqrt{-1}\not\in k\) or the rational prime \((2)\) has at least two extensions in \(k\). In addition, \({\text{PSL}}(2,11)\) is \(k\)-admissible over \(\mathbb{Q}\) and over any number field \(k\) with at least two extensions of the dyadic prime. The proofs are based on multi-parameter polynomials with Galois group \({\text{PSL}}(2,7)\), resp. \({\text{PSL}}(2,11)\) given by G. Malle.

MSC:

12F12 Inverse Galois theory
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References:

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