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
Famous quotes containing the words product and/or fields:
“The UN is not just a product of do-gooders. It is harshly real. The day will come when men will see the UN and what it means clearly. Everything will be all right—you know when? When people, just people, stop thinking of the United Nations as a weird Picasso abstraction, and see it as a drawing they made themselves.”
—Dag Hammarskjöld (1905–1961)
“Smart lad, to slip betimes away
From fields where glory does not stay,
And early though the laurel grows
It withers quicker than the rose.”
—A.E. (Alfred Edward)