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 ℚ of finite degree n, is always a product of fields isomorphic to ℝ or ℂ. The totally real number fields are those for which only real fields occur: in general there are r1 real and r2 complex fields, with r1 + 2r2 = 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 where ℚp is the field of p-adic numbers. This is a product of finite extensions of ℚp, in 1–1 correspondence with the completions of K for extensions of the p-adic metric on ℚ.
Read more about this topic: Tensor Product Of Fields
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