Simple algebras over local fields

Abstract

Let D be a division algebra of finite dimension over any field K; we will consider left vector-spaces over D, whose dimension will always be assumed finite and <0. If V and W are such spaces, we write Hom(V,W) for the space of homomorphisms of V into W _, and let it operate on the right on V; in other words, if a is such a homo-morphism, and v ∈ V, we write v α for the image of v under α. We consider Hom(V, W), in an obvious manner, as a vector-space over K; as such, it has a finite dimension, since it is a subspace of the space of K-linear mappings of V into W. As usual, we write End (V) for Hom(V, V).