\(k\)-admissibility of certain symmetric groups over algebraic number fields (Q1908476)

Let \(k\) be an algebraic number field and \(G\) a finite group. Then \(G\) is said to be \(k\)-admissible if there exists a Galois extension \(L/k\) with Galois group \(G\) which is a maximal subfield of a \(k\)-division ring, that is, of a division ring which has \(k\) as its center. The author proves necessary and sufficient conditions for an algebraic number field \(k\) over which the symmetric groups \(S_n\) are \(k\)-admissible, for values of \(n\) from 6 to 11, as well as for 16 and 17. The author's main result is as follows: Let \(k\) be an algebraic number field. Then the groups \(S_6\), \(S_7\), and \(S_8\) are \(k\)-admissible if and only if the prime 2 splits in \(k\), the groups \(S_9\), \(S_{10}\), and \(S_{11}\) are \(k\)-admissible if and only if the primes 2 and 3 split in \(k\), and the groups \(S_{16}\) and \(S_{17}\) are \(k\)-admissible if and only if the primes 2, 3, and 5 split in \(k\).