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## Main Question or Discussion Point

There are equivilant definitions of Galois extensions listed here http://mathworld.wolfram.com/GaloisExtensionField.html but I'm confused about the equivilence of 1 and 2.

What am I doing wrong here? Take [tex]K[/tex] to be the splitting field of [tex]X^4-2[/tex] over [tex]\mathbb{Q}[/tex]. This is exactly property 1. But if you consider the automorphism of complex conjugation then it fixes the intermediate field [tex]\mathbb{Q} \subset (K\cap \mathbb{R})\subset K [/tex] which contradicts property 2. (yes I am abusing notation slightly. just consider some embedding of [tex]K[/tex] in [tex]\mathbb{C}[/tex] and my intersection makes sense)

I assume I've made a mistake since it's been a while since I've checked these sorts basic properties but where?

What am I doing wrong here? Take [tex]K[/tex] to be the splitting field of [tex]X^4-2[/tex] over [tex]\mathbb{Q}[/tex]. This is exactly property 1. But if you consider the automorphism of complex conjugation then it fixes the intermediate field [tex]\mathbb{Q} \subset (K\cap \mathbb{R})\subset K [/tex] which contradicts property 2. (yes I am abusing notation slightly. just consider some embedding of [tex]K[/tex] in [tex]\mathbb{C}[/tex] and my intersection makes sense)

I assume I've made a mistake since it's been a while since I've checked these sorts basic properties but where?