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Galois extension of a field with Characteristic 0

  1. Aug 6, 2007 #1
    This is something I've been trying to work on on my own for the past few days but I'm not sure how to approach it.

    My Question:
    a. Let E be a Galois extension of a field F with characteristic 0. Prove that there is a unique smallest subfield K such that [itex] F \subseteq K \subseteq E [/itex], K is normal over F and E is subradical over K.
    [One needs the following result: Let [itex]A[/itex] and [itex]B[/itex] be solvable subgroups of a group [itex]G[/itex] and suppose that [itex]A[/itex] is normal in [itex]G[/itex]. Then [itex]AB[/itex] is solvable.]

    b. Let f be an irreducible polynomial over [itex]\mathbb{Q}[/itex] which has degree 5 and at least two complex roots. Prove that [itex]Gal(f)[/itex] has order 10, 20, 60, or 120.

    Thanks in advance for any kind of direction that you can provide me with...
     
  2. jcsd
  3. Aug 7, 2007 #2

    mathwonk

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    what does smallest mean? the intersectiuoin of all such thigns? or smallest degree?

    and are your extensions of finite degree?

    so i guess you are claiming there is a unique largest normal solvable subgroup of a group. your description majkes it fairly obvious how to proceed.
     
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