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:
“Good is a product of the ethical and spiritual artistry of individuals; it cannot be mass-produced.”
—Aldous Huxley (1894–1963)
“The world’s great day is growing late,
Yet strange these fields that we have planted
So long with crops of love and hate.”
—Edwin Muir (1887–1959)