Proposition 10

Let G be a finite group. Let L be a Galois algebra over the field K with group G, and let $K'$ be an extension of K. Then the $(K',G)$-algebra $L_{(K')}$ is a Galois algebra over $K'$.

We use property (v) in Theorem 2 by observing that if the K-algebra L is étale, then the $K'$-algebra $L_{(K')}$ is too (V, §6, No. 5, p. 32, Corollary 2), that we have the equality $[L_{(K')}: K'] = [L : K]$, and that the ring of invariants of G in $L_{(K')}$ is $(L^G)_{(K')}$, where $L^G$ is the ring of invariants of G in L.