Exercise 2, § 11

a) Let E be a splitting extension of the polynomial $X^3 - 2 \in Q[X]$. The field E is a cyclic extension of degree 3 of $R_3(Q)$ but contains no cyclic extension of degree 3 of Q. b) The field $E = R_9(Q)$ is a cyclic extension of degree 3 of $R_3(Q)$ and contains a cyclic extension of degree 3 of Q. c) With the hypothesis and notation as in V, p. 154, Ex. 7, show that for $K(\alpha, \beta)$ to be Galois over K it is necessary and sufficient for d to be a square in $K(\alpha)$; for $K(\beta)$ to be cyclic over K it is necessary and sufficient for d to be a square in K.