Remark

Let L be a Galois extension of the field K of finite degree, and let G be its Galois group; we denote the identity mapping on G by $\lambda$. Then L endowed with $\lambda$ is a Galois algebra over K with group G. Let $K'$ be an extension of K. By Proposition 10, the algebra $L_{(K')}$ is Galois over $K'$, but it is not in general an extension of $K'$. Analogously, by Proposition 11, the tensor product of Galois extensions E and F of K of finite degree can be viewed as a Galois algebra; in general, it is not a Galois extension of K. It is one, however, if E and F are moreover linearly disjoint subextensions of an extension of K (V, §2, No. 5, p. 13 and V, §10, No. 1, p. 57, Proposition 1).