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:
“A product of the untalented, sold by the unprincipled to the utterly bewildered.”
—Al Capp (19091979)
“Or seen the furrows shine but late upturned,
And where the fieldfare followed in the rear,
When all the fields around lay bound and hoar
Beneath a thick integument of snow.
So by Gods cheap economy made rich
To go upon my winters task again.”
—Henry David Thoreau (18171862)