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Galois Extension field properties

  1. Aug 24, 2005 #1
    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?
  2. jcsd
  3. Aug 24, 2005 #2


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    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!
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