Simple algebras over A-fields

Abstract

In this Chapter, k will be an A-field; we use all the notations introduced for such fields in earlier Chapters, such as k _A , k _v , r _v , etc. We shall be principally concerned with a simple algebra A over k; as stipulated in Chapter IX, it is always understood that A is central, i. e. that its center is k, and that it has a finite dimension over k; by corollary 3 of prop. 3, Chap. IX–1, this dimension can then be written as n², where n is an integer ⩾1. We use A _v , as explained in Chap­ters III and IV, for the algebra A _v = A⊗k _v over k _v , where, in agreement with Chapter IX, it is understood that the tensor-product is taken over k. By corollary 1 of prop. 3, Chap. IX–1, this is a simple algebra over k _v ; therefore, by th. 1 of Chap. IX–1, it is isomorphic to an algebra M _m(v) (D(v)) where D(v) is a division algebra over k _v ; the dimension of D(v) over k _v can then be written as d(v) ², and we have m(v)d(v) = n; the algebra D(v) is uniquely determined up to an isomorphism, and m(v) and d(v) are uniquely determined. One says that A is unramified or ramified at v according as A _v is trivial over k _v or not, i. e. according as d(v) = 1 or d(v)>1.