Proposition 9

Let G be a finite group, H be a subgroup of G, and E be a Galois algebra over the field K with group H. Then the $(K,G)$-algebra Coind$^G_H(E)$ deduced from E by coinduction from H to G is a Galois algebra over the field K with group G.

Since E is a Galois algebra over K, by property (v) of Theorem 2 of VIII, p. 308, it is reduced, we have Card(H) = [E : K], and K is the ring of invariants of H in E. But by Lemma 7 of VIII, p. 305, the algebra F = Coind$^G_H(E)$ is reduced, and we have

[F : K] = (G : H)[E : K] = (G : H) Card(H) = Card(G),

and K is the ring of invariants of G in F. So F is a Galois algebra by the criterion given in Theorem 2, (v).