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Galois extension

  1. Feb 22, 2010 #1
    This stuff is killing me...

    Let [tex]K \leq M \leq L[/tex] be fields such that L is galois over M and M is galois over K. We can extend [tex]\phi \in G(M/K)[/tex] to an automorphism of L to show L is galois over K.

    I need help filling in the details in why exactly L is galois over K.
     
  2. jcsd
  3. Feb 23, 2010 #2
    Something like this, I think: let [tex][M:K]=a[/tex] and [tex][L:M]=b[/tex]. Since [tex]M/K[/tex] is Galois, there are [tex]a[/tex] automorphisms of [tex]M[/tex] that fix [tex]K[/tex]. For the same reason, there are [tex]b[/tex] automorphisms of [tex]L[/tex] that fix [tex]M[/tex]. So there are [tex]ad[/tex] automorphisms of [tex]L[/tex] that fix [tex]K[/tex]. Since [tex]ad[/tex] is also the degree of [tex]L/K[/tex], it's Galois.

    If you're using Dummit and Foote, check out theorems 13.8 and 13.27.
     
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