Dihedral crossed products of exponent 2 are Abelian. (Q1401598)

The author considers Galois splitting fields of central simple algebras of exponent~\(2\). For an arbitrary base field \(F\), a finite group \(G\) is said to split a central simple \(F\)-algebra \(A\) if there is a Galois extension of \(F\) with Galois group isomorphic to a subgroup of \(G\) which splits \(A\). Of special interest is the case where \(G\) is a cyclic group \(C_{n}\). The main result of this short note states that for any integer \(m\) and any multiple \(n\) of \(m\), every central simple \(F\)-algebra of exponent~2 which is split by the dihedral group \(D_{n}\) of order~\(2n\) is also split by a direct product \(C_{n/m}\times D_{m}\), provided \(F\) contains a primitive \((n/m)\)-th root of unity. In particular (for \(m=1\)), central simple algebras of exponent~\(2\) split by a dihedral group are also split by an Abelian group if the center contains enough roots of unity.