Basic Number Theory pp 202-212 | Cite as

# 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

*is central, i. e. that its center is*

**A***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

*is*

**A***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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