In mathematics, algebras A, B over a field k inside some field extension \Omega of k are said to be linearly disjoint over k if the following equivalent conditions are met: Note that, since every subalgebra of \Omega is a domain, (i) implies is a domain (in particular reduced). 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. However, there are examples where 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 \Omega generated by A, B, resp. are linearly disjoint over k. (cf. Tensor product of fields) Suppose A, B are linearly disjoint over k. If, are subalgebras, then A' and B' 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.)