Exercise 1, § 9

The polynomial $X^2 - 2$ is irreducible in $\mathbf{Q}[X]$ (cf. I, p. 51, Th. 7); let $a$ be one of its roots in an algebraic closure $\Omega$ of $\mathbf{Q}$. Show that the polynomial $X^2 - \alpha$ is irreducible over $E = \mathbf{Q}(a)$; let $\beta$ be one of the roots of this polynomial in $\Omega$ and let $F = E(\beta) = \mathbf{Q}(\beta)$. Show that $F$ is not a quasi-Galois extension of $\mathbf{Q}$; what is the quasi-Galois extension of $\mathbf{Q}$ generated by $F$?