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

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