Subfields and Splitting Fields of Division Algebras

In this chapter we apply the machinery developed in previous chapters to analyze the subfields and splitting fields of division algebras over a Henselian field F. In §9.1 we give properties of the splitting fields of tame division algebra D with center F, with particularly strong criteria proved if D is inertial or totally ramified over F. This leads to explicit constructions of several interesting examples of division algebras, including noncyclic division algebras of degree p 2 with no maximal subfield of the form $F(\!\sqrt[p^{2}]{a})$ in Examples 9.15, 9.17, and 9.18; noncyclic p-algebras in Ex. 9.26; noncrossed product algebras including universal division algebras in Th. 9.30 and division algebras over Laurent series over $\mathbb {Q}$ , noncrossed products whose degree exceeds the exponent in Cor. 9.46; and crossed product division algebras with only one Galois group for all maximal subfields Galois over the center in Prop. 9.28[9.28].