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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1967 Springer-Verlag Berlin · Heidelberg
About this chapter
Cite this chapter
Weil, A. (1967). Simple algebras over A-fields. In: Basic Number Theory. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, vol 144. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-00046-5_11
Download citation
DOI: https://doi.org/10.1007/978-3-662-00046-5_11
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-00048-9
Online ISBN: 978-3-662-00046-5
eBook Packages: Springer Book Archive