Joined: Mon Nov 09, 2015 11:52 am Posts: 77 Location: Limassol/Pyla Cyprus

We observe that complex conjugation is a field automorphism of \(\mathbb{K}\). If \(f(x)\) has a nonreal root, then \(\mathbb{K}\) has a nonreal element and therefore this automorphism has order \(2\). Let \(\mathbb{L}\) be the fixed field of the automorphism. (I.e. \(\mathbb{L}=\mathbb{K} \cap \mathbb{R}\).) By the fundamental theorem of Galois theory we have that \([\mathbb{K}:\mathbb{L}]=2\) and so \([\mathbb{K}:\mathbb{Q}]=[\mathbb{K}:\mathbb{L}][\mathbb{L}:\mathbb{Q}]\) is even, a contradiction.

