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Math Help - Cyclotomic field

  1. #1
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    Cyclotomic field

    Show that the union of \mathbb Q_{2^n}\cap \mathbb R, where n is from 3 to  \infty , is a (profinite) normal extension of \mathbb Q.
    And show that its Galois group is the additive group of the integer 2-adic number.
    Last edited by ZetaX; April 20th 2009 at 09:18 AM.
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  2. #2
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    CAn somebody help me with this?
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  3. #3
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    Quote Originally Posted by ZetaX View Post

    Show that the union of \mathbb Q_{2^n}\cap \mathbb R, where n is from 3 to  \infty , is a (profinite) normal extension of \mathbb Q.
    And show that its Galois group is the additive group of the integer 2-adic number.
    well, K=\bigcup (\mathbb{Q}_{2^n} \cap \mathbb{R}) is normal because \mathbb{Q}_{2^n}/\mathbb{Q} is abelian and thus each K_n=\mathbb{Q}_{2^n} \cap \mathbb{R} is normal. i'm not expert in infinite Galois theory but i'm pretty sure the key to the second part of your

    problem is this isomorphism: \text{Gal}(K/\mathbb{Q}) \cong \varprojlim \text{Gal}(K_n/\mathbb{Q}). now the question is what exactly is \text{Gal}(K_n/\mathbb{Q}) ?
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