Exercise 18, § 14

Let K be a commutative field, Z a cyclic extension of K (V, §11, No. 6, p. 85) of degree $n\geqslant 3$, and $\sigma$ a generator of the Galois group of Z over K. Denote by A the ring $\mathbf{M}_2(Z)$ and by B the subring of A consisting of the matrices $\begin{pmatrix} z & 0 \\ 0 & \sigma(z) \end{pmatrix}$ for $z\in Z$. a) Keep the notation of Exercise 17. Prove that there exists a Galois group G of automorphisms of A such that $A^G= K, (G : G_0) =n$, and $U_0(G) = A$.

b) Prove that B is a simple subring of A but that its commutant $B'_0$ in A is isomorphic to $Z\times Z$.

c) Let H be the group of automorphisms of A fixing B. Prove that $A^H$ is equal to $B'_0$. Construct an isomorphism from B to Z that fixes K but does not extend to an automorphism of A (cf. Exercise 17, b)).

d) Prove that the subring B of A is weakly Galois but not Galois (Exercise 16; describe the commutant of B in Ω).