Fundamental theorem of algebra

9.4.1 Lemma. Let F a field of characteristic zero such that every finite extension K of F, K ≠ F such that [K: F] is divisible by a prime number p. Then every finite extension of F has degree a power of p.

Proof. Let K a finite extension of F. It's not restrictive to suppose that K is a Galois extension of F. Let G(K, F) the Galois group of such extension. Since [K: F] by hypothesis is divisible by p, and [K: F] = |G(K,F)|, if |G(K,F)| is divisible by p^α, but not by p^α+1, then, by the First Sylow theorems, G(K,F) contains a subgroup H of order p^α. The field fixed by K_H is such that

[K_{H} : F] = \frac{| G (K, F) |}{| G (K, K_{H}) |} = \frac{| G (K, F) |}{| H |}

and [K: F] is not divisible by p, hence it must be [K: F] = 1, i.e. K_H = F, and consequently G(K,F) = H, thus [K:F] = |G(K,F)| = pⁿ for some n. □

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