 Solution

Eisenstein's criterion says that an integer polynomial with leading coefficient not divisible by a prime $\ell$, all other coefficients divisible by $\ell$, and constant coefficient not divisible by ${\ell }^2$ is irreducible over the rationals, provided it is primitive; the usual Gauss's lemma supplies the passage from integer to rational factors.

Apply it with $\ell =2$ toAll nonleading coefficients are even and the constant is not divisible by four. A translation of the variable preserves reducibility, so $x^8+1$ is irreducible over $\mathbb{Q}$.

Let $\zeta =e^{i\pi /8}$ and $c=\cos (\pi /8)$. The minimal polynomial of $\zeta$ is $x^8+1$, so $[\mathbb{Q}(\zeta ):\mathbb{Q}]=8$. Since $2c=\zeta +{\zeta }^{-1}$, the number $\zeta$ satisfies $t^2-2ct+1=0$ over $\mathbb{Q}(c)$. This real field cannot contain the nonreal $\zeta$, so that extension has degree exactly two. The tower law for field extensions gives