# Galois Extension field properties

1. Aug 24, 2005

### snoble

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 $$K$$ to be the splitting field of $$X^4-2$$ over $$\mathbb{Q}$$. This is exactly property 1. But if you consider the automorphism of complex conjugation then it fixes the intermediate field $$\mathbb{Q} \subset (K\cap \mathbb{R})\subset K$$ which contradicts property 2. (yes I am abusing notation slightly. just consider some embedding of $$K$$ in $$\mathbb{C}$$ 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?

2. Aug 24, 2005

### Hurkyl

Staff Emeritus
I think it means the collection of automorphisms as a whole, not individually.

For any extension field E of F, note that the identity is an automorphism fixing E, F, and all fields in-between!