## There are no fields between QQ and QQ(¹³√2)

Let $\alpha$ be a root of $p(x) = x^{13} - 2$. Does there exist a field $K$ such that $\mathbb{Q} \subsetneq K \subsetneq \mathbb{Q}(\alpha)$?

Note that $p(x)$ is irreducible by Eisenstein’s criterion; hence $\mathsf{dim}_\mathbb{Q} \mathbb{Q}(\alpha) = 13$. Suppose $K$ is a field in between $\mathbb{Q}$ and $\mathbb{Q}(\alpha)$. By Theorem 5.7 in TAN, we have $\mathsf{dim}_\mathbb{Q} \mathbb{Q}(\alpha) = \mathsf{dim}_\mathbb{Q} F \cdot \mathsf{dim}_F \mathbb{Q}(\alpha)$. Since 13 is prime, either $\mathsf{dim}_\mathbb{Q} F = 1$ or $\mathsf{dim}_F \mathbb{Q}(\alpha) = 1$; so $F = \mathbb{Q}$ or $F = \mathbb{Q}(\alpha)$.

Thus no fields intermediate to $\mathbb{Q}$ and $\mathbb{Q}(\alpha)$ exist.