\(K\)-admissibility of \(S_{12}, S_{13}, S_{14}, S_{15}\) (Q1125903)

Let \(k\) be a field and \(G\) a finite group. Then \(G\) is said to be \(k\)-admissible if there exists a Galois extension \(L/k\) such that \(\text{Gal}(L/k)\cong G\) and \(L\) is a maximal subfield of a finite-dimensional \(k\)-central division ring, i.e. if there is a \(k\)-division ring which is a crossed product for \(G\). The notion of \(k\)-admissibility was introduced by \textit{M. Schacher} [J. Algebra 9, 451-477 (1968; Zbl 0174.34103)], where it was shown that every finite group is \(k\)-admissible for some number field \(k\), and that for a given number field \(k\), only finitely many symmetric groups are \(k\)-admissible. This paper gives complete characterizations of those number fields \(k\) for which the symmetric groups \(S_{12}\), \(S_{13}\), \(S_{14}\), and \(S_{15}\) are \(k\)-admissible. The characterizations are based on the divisors in \(k\) of the primes 2, 3, and 5. This completes the characterization of fields over which \(S_n\) is admissible for \(n\leq 17\), as Schacher's original article answers the question for \(n=2,3\), while the cases \(n=4,5\) have been addressed by \textit{S. Liedahl} [J. Algebra 169, 965-983 (1994; Zbl 0820.20024)], and \(6\leq n\leq 11\) and \(n=16,17\) have been considered by \textit{Z. Girnius} [J. Algebra 177, 277-296 (1995; Zbl 0848.12003)].