Galois splitting fields of a universal division algebra (Q1178058)

Let \(UD(k,n)\) denote the universal division algebra of degree \(n\) (dimension \(n^ 2\)) over a field \(k\), generated by \(m\) \((\geq 2)\) generic matrices of order \(n\). In an epoch-making paper of 1972, the author proved that \(UD(\mathbb{Q},n)\) does not contain any maximal subfield which is Galois over the center if \(n\) is divisible by 8 or by the square of an odd prime. The present paper gives quantitative improvements of this result, by investigating the minimal degree of a Galois splitting field of \(UD(k,n)\). Using the primary decomposition of division algebras, it suffices to consider the case where \(n\) is a prime power. If \(N\) is an integer such that \(p^ N\) divides the degree of a Galois splitting field of \(UD(k,p^ n)\), then the main result of the paper yields an explicit lower bound for \(N\) in terms of \(p\) and \(n\). This lower bound is of the same order of magnitude as \((n/2)\log p\) for \(n\) large. This result is a refinement of those obtained by the author and the reviewer in a series of papers [J. Algebra 98, 95-101 (1986; Zbl 0588.16011); Isr. J. Math. 50, 114-144 (1985; Zbl 0564.16015); ibid. 54, 266-290 (1986; Zbl 0599.20086)]. The present paper also fills a gap in the literature by providing a detailed proof of a fundamental result which had been announced previously and was used in the above-mentioned series of papers: If \(UD(k,n)\) is split by a Galois extension with Galois group \(G\), then every division algebra of degree \(n\) whose center contains \(k\) is split by a Galois extension with a Galois group isomorphic to a subgroup of \(G\).