Allman, Elizabeth S.; Schacher, Murray M. Division algebras with \(\text{PSL} (2,q)\)-Galois maximal subfields. (English) Zbl 0977.12004 J. Algebra 240, No. 2, 808-821 (2001). 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. Reviewer: M.Epkenhans (Paderborn) Cited in 1 ReviewCited in 3 Documents MSC: 12F12 Inverse Galois theory Keywords:division algebras; admissable extensions; Galois theory PDFBibTeX XMLCite \textit{E. S. Allman} and \textit{M. M. Schacher}, J. Algebra 240, No. 2, 808--821 (2001; Zbl 0977.12004) Full Text: DOI References: [1] Erbach, D.; Fisher, J.; McKay, J., Polynomials with PSL(2,7) as Galois group, J. Number Theory, 11, 69-75 (1979) · Zbl 0405.12011 [2] Fein, B.; Saltman, D.; Schacher, M., Crossed products over rational function fields, J. Algebra, 156, 454-493 (1993) · Zbl 0793.16017 [3] Feit, P.; Feit, W., The \(K\)-admissibility of SL(2,5), Geom. Dedicata, 36, 1-13 (1990) · Zbl 0702.12001 [4] Kaplansky, I., Fields and Rings. Fields and Rings, Chicago Lectures in Mathematics (1969), Univ. of Chicago Press: Univ. of Chicago Press Chicago [5] LaMacchia, S., Polynomials with Galois group PSL(2,7), Comm. Algebra, 8, 983-992 (1980) · Zbl 0436.12005 [6] LaMacchia, S., Polynomials with Galois group PSL(2,11), Comm. Algebra, 9, 613-625 (1981) · Zbl 0464.12012 [7] G. Malle, Some multi-parameter polynomials with given Galois group, J. Symbolic Comput, to appear.; G. Malle, Some multi-parameter polynomials with given Galois group, J. Symbolic Comput, to appear. · Zbl 0967.12005 [8] Malle, G.; Matzat, B. H., Realisierung von Gruppen \(PSL_2\)(\(F_p\)) als Galois-gruppen uber \(Q\), Math. Ann., 272, 549-565 (1985) · Zbl 0559.12005 [9] Malle, G.; Matzat, B. H., Inverse Galois Theory (1999), Springer-Verlag: Springer-Verlag Berlin · Zbl 0940.12001 [10] Saltman, D., Generic Galois extensions and problems in field theory, Adv. in Math., 43, 250-283 (1983) · Zbl 0484.12004 [11] Schacher, M., Subfields of division rings, I, J. Algebra, 9, 451-477 (1968) · Zbl 0174.34103 [12] Schacher, M.; Sonn, J., \(k\)-admissibility of \(A_6\) and \(A_7\), J. Algebra, 145, 333-338 (1992) · Zbl 0739.12004 [13] Smith, G. W., Some polynomials over \(Q(t)\) and their Galois groups, Math. Comp., 69, 775-796 (2000) · Zbl 0962.11041 [14] W. Trinks, Ein Beispiel eines Zahlkorpers mit der Galoisgruppe PGL(3,\(Q\); W. Trinks, Ein Beispiel eines Zahlkorpers mit der Galoisgruppe PGL(3,\(Q\) [15] Weiss, E., Algebraic Number Theory (1963), Chelsea: Chelsea New York 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.