How often can a finite group be realized as a Galois group over a field?
Let \(G\) be a finite group and let \(K\) be a field. Then \(\nu (G,K)\) denotes the number of Galois extensions \(L/K\) inside a fixed separable closure with Galois group \(G(L/K)\simeq G\). For a class \({\mathcal C}\) of fields the realization multiplicity \(\nu_{\mathcal C}(G)\) is defined to be the infimum of all \(\nu(G,K)\), \(K\) running through all fields \(K\) in \({\mathcal C}\) with \(\nu(G,K) >0\). The absolute realization multiplicity is obtained by choosing \({\mathcal C}\) the class of all fields. For \(G\) abelian and \({\mathcal C}\) the class of algebraic extensions of \(\mathbb{Q}\) an explicit formula for \(\nu_{\mathcal C}(G)\) is given. The authors further treat the case of an abelian \(p\)-group and the class \({\mathcal E}_p\) which is conjectured to be the class of all fields of characteristic \(\neq p\) for which the Galois group of the maximal \(p\)-extension is finitely generated. The paper further contains some partial results on non-abelian groups, for example non-abelian groups of order \(2^n\) and exponent \(2^{n-1}\), non-abelian groups of order 16 and some direct products of 2-groups.
scientific article; zbMATH DE number 1334553