\(\mathbb{Q}\)-adequacy of Galois 2-extensions (Q696274)

A finite field extension \(L\) of \(F\) is called \(F\)-adequate if there exists an \( F\)-central division algebra \(D\) containing \(L\) as a maximal commutative subfield. By a first result, a Galois \(2\)-extension \(L\) of \({\mathbb{Q}}\) is \({\mathbb{Q}} \)-adequate if and only if the maximal elementary abelian extension \(K\) of \({\mathbb{Q}}\) inside \(L\) has degree 2 or 4 over \({\mathbb{Q}}\). In chapter 3 of the paper under review the authors give equivalent conditions of a biquadratic extension \({\mathbb{Q}}(\sqrt m,\sqrt n)/{\mathbb{Q}}\) to be \({\mathbb{Q}}\)-adequate. These conditions are in terms of Legendre symbols and congruence relations \(\mod 8\). These results are based on the fact, that such an extension is \({\mathbb{Q}}\)-adequate if and only if \([{\mathbb{Q}}_p(\sqrt m,\sqrt n):{\mathbb{Q}}_p]=4\) for two different rational primes \(p\).