Tensor Product of Fields

The tensor product of fields is the best available construction on fields with which to discuss all the phenomena arising. As a ring, it is sometimes a field, and often a direct product of fields; it can, though, contain non-zero nilpotents (see radical of a ring).

If K and L do not have isomorphic prime fields, or in other words they have different characteristics, they have no possibility of being common subfields of a field M. Correspondingly their tensor product will in that case be the trivial ring (collapse of the construction to nothing of interest).

Read more about Tensor Product Of Fields:  Compositum of Fields, The Tensor Product As Ring, Analysis of The Ring Structure, Examples, Classical Theory of Real and Complex Embeddings, Consequences For Galois Theory

Other articles related to "tensor product of fields":

Tensor Product Of Fields - Consequences For Galois Theory
... This gives a general picture, and indeed a way of developing Galois theory (along lines exploited in Grothendieck's Galois theory) ... It can be shown that for separable extensions the radical is always {0} therefore the Galois theory case is the semisimple one, of products of fields alone ...

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