Exercise 1, § 2

a) Show that for every integer $a \neq 0$ the polynomial $X^4 - aX - 1$ is irreducible in $\mathbf{Q}[X]$. b) Let $a$ be a root of $X^4 - aX - 1$ in an extension of $\mathbf{Q}$, and let $K = \mathbf{Q}(\alpha)$, which is of degree 4 over $\mathbf{Q}$. Show that there is an infinity of values of $a \in \mathbf{Z}$ such that $K$ contains no subfield $F$ distinct from $\mathbf{Q}$ and $K$.