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The tensor product of fields $L{\otimes }_{K}M$ is a $K$-algebra, not necessarily a field. If $L$ and $M$ sit in one overfield, the map $l\otimes m\mapsto lm$ has an image that generates their compositum as a field. When the extensions are algebraic, this image is already a field. Injectivity of this multiplication map is equivalent to linear disjointness.