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.

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