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Math Help - Non-algebraic Galois Extension

  1. #1
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    Non-algebraic Galois Extension

    Say E/F is Galois*.
    Must it be the case that E/F is algebraic?
    (This was not addressed in my book).



    *)I just realized how bad my question is. A Galois extension E/F is an algebraic extension such that F = E^{\text{Gal}(E/F)}. Note we use "algebraic" within the definition itself. My question would be more appropriately asked if F = E^{\text{Gal}(E/F)} then must it follow that E/F is algebraic?
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  2. #2
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    Quote Originally Posted by ThePerfectHacker View Post
    My question would be more appropriately asked if F = E^{\text{Gal}(E/F)} then must it follow that E/F is algebraic?
    this is a good question! the answer is No! because, for example, if x is transcendental over F and E=F(x), then

    it can be proved (not very easily though!) that E^{\text{Gal}(E/F)}=F. the main part of the proof is to show that \text{Gal}(E/F)

    consists of all maps \sigma defined by \sigma(f(x))=f(\frac{ax+b}{cx+d}), \ \forall f \in E, where a,b,c,d \in F and ad - bc \neq 0.
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  3. #3
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    Quote Originally Posted by NonCommAlg View Post
    this is a good question! the answer is No! because, for example, if x is transcendental over F and E=F(x), then

    it can be proved (not very easily though!) that E^{\text{Gal}(E/F)}=F. the main part of the proof is to show that \text{Gal}(E/F)

    consists of all maps \sigma defined by \sigma(f(x))=f(\frac{ax+b}{cx+d}), \ \forall f \in E, where a,b,c,d \in F and ad - bc \neq 0.
    I actually was thinking about \mathbb{Q}(\pi)/\mathbb{Q}, but I never did any of the details to prove it is Galois.

    Now this leads me to my next question: is there anything interesting about a non-algebraic Galois extension?
    (Maybe my books talk about this stuff near the end on transcendental extensions,
    but it would be interesting to me to know ahead of time).
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    Quote Originally Posted by ThePerfectHacker View Post
    Now this leads me to my next question: is there anything interesting about a non-algebraic Galois extension?
    i'm not sure, probably it's more complicated than interesting! for example we know that in algebraic Galois extensions,

    if E/F is Galois and K is a subfield of E which contains F, then E/K would be Galois too. now in order to have this nice

    property in non-algebraic extensions we may define that E/F to be Galois if E^{\text{Gal}(E/K)}=K, for any subfield K of E which

    contains F. but then we'd have a problem: with this definition there is no non-algebraic Galois extension of any field of

    characteristic p > 0 (this is easy to prove!)
    Last edited by NonCommAlg; May 28th 2008 at 09:19 PM.
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