Linearly disjoint

of k are said to be linearly disjoint over k if the following equivalent conditions are met: Note that, since every subalgebra of

Conversely if A and B are fields and either A or B is an algebraic extension of k and

is a domain then it is a field and A and B are linearly disjoint.

is a domain but A and B are not linearly disjoint: for example, A = B = k(t), the field of rational functions over k. One also has: A, B are linearly disjoint over k if and only if the subfields of

are linearly disjoint over k. (cf.

Tensor product of fields) Suppose A, B are linearly disjoint over k. If

are linearly disjoint over k. Conversely, if any finitely generated subalgebras of algebras A, B are linearly disjoint, then A, B are linearly disjoint (since the condition involves only finite sets of elements.)

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