Maximal subfields of \(\mathbb{Q} (i)\)-division rings (Q1816538)

We determine the \(\mathbb{Q} (i)\)-division rings which have maximal subfields of the form \(E(i)\), where \(E/\mathbb{Q}\) is cyclic. These are precisely the \(\mathbb{Q} (i)\)-division rings having maximal subfields which are abelian over \(\mathbb{Q}\). More generally, we determine the \(\mathbb{Q} (i)\)-division rings having maximal subfields which are Galois over \(\mathbb{Q}\). We show that the existence of such subfields in a \(\mathbb{Q} (i)\)-division ring \(D\) is determined by the 2-part of the Sylow decomposition of \(D\).