Frattini closed groups and adequate extensions of global fields. (Q696273)

The authors consider the following problem: If \(F\) is a field and \(L\) a finite extension of \(F\), does there exist an \(F\)-central division algebra \(D\) containing \(L\) as a maximal commutative subfield? If there exists such a \(D\), \(L\) is said to be \(F\)-adequate. In the paper they give a certain group-theoretic condition and show that if a finite Galois extension \(L\) of a global field \(F\) has a Galois group \(G\) satisfying this condition, then the \(F\)-adequacy of \(L\) is determined by the \(F\)-adequacy of the subfield fixed by the Frattini subgroup of \(G\). In the final part of the paper they show that a class of groups that includes finite supersolvable groups satisfies this condition and exhibit some groups that do not satisfy it.