Simple algebras over A-fields
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 n 2 , 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.
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