Editing Tensor product of fields (section)

==Classical theory of real and complex embeddings==

In [[algebraic number theory]], tensor products of fields are (implicitly, often) a basic tool. If ''K'' is an extension of <math>\mathbb{Q}</math> of finite degree ''n'', <math>K\otimes_{\mathbb Q}\mathbb R</math> is always a product of fields isomorphic to <math>\mathbb{R}</math> or <math>\mathbb{C}</math>. The [[totally real number field]]s are those for which only [[real number|real]] fields occur: in general there are ''r''<sub>1</sub> real and ''r''<sub>2</sub> complex fields, with ''r''<sub>1</sub>&thinsp;+&nbsp;2''r''<sub>2</sub> = ''n'' as one sees by counting dimensions. The field factors are in 1–1 correspondence with the ''real embeddings'', and ''pairs of complex conjugate embeddings'', described in the classical literature.

This idea applies also to <math>K\otimes_{\mathbb Q}\mathbb Q_p,</math> where <math>\mathbb{Q}</math><sub>''p''</sub> is the field of [[p-adic number|''p''-adic numbers]]. This is a product of finite extensions of <math>\mathbb{Q}</math><sub>''p''</sub>, in 1–1 correspondence with the completions of ''K'' for extensions of the ''p''-adic metric on <math>\mathbb{Q}</math>.