Exercise 12, § 10

Let K be an infinite field, N a Galois extension of degree $n$ of K, $f \in K[X]$ the minimal polynomial of an element 0 of N such that $N = K(0)$. The group $\Gamma = \mathrm{Gal}(N/K)$ is allowed to operate on the polynomials in $N[X]$ by operating on the coefficients of these polynomials. a) Let $g \in N[X]$ be the polynomial of degree $\leq n - 1$ such that $g(\theta) = 1, g(\sigma(\theta)) = 0$ for all $\sigma \neq 1$ in $\Gamma$. For every subgroup $A$ of $\Gamma$ let $p_A \in N[X]$ be the polynomial $\det(\sigma \tau(g))_{\sigma \in A, \tau \in A}$. Show that

and deduce that $p_A \neq 0$. b) Deduce that there exists an element $a \in K$ such that for every subgroup $A$ of $\Gamma$ the conjugates of $g(a)$ over the subfield $k(\Delta)$ form a normal basis of N over $k(\Delta)$.