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:
“...In the past, as now, [Hollywood] was a stamping ground for tastelessness, violence, and hyperbole, but once upon a time it turned out a product which sweetened the flavor of life all over the world.”
—Anita Loos (18881981)
“Earth has not anything to show more fair:
Dull would he be of soul who could pass by
A sight so touching in its majesty:
This city now doth, like a garment, wear
The beauty of the morning; silent, bare,
Ships, towers, domes, theatres and temples lie
Open unto the fields and to the sky;
All bright and glittering in the smokeless air.”
—William Wordsworth (17701850)