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Thread: Cyclotomic field

  1. #1
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    Cyclotomic field

    Show that the union of $\displaystyle \mathbb Q_{2^n}\cap \mathbb R$, where $\displaystyle n$ is from $\displaystyle 3$ to $\displaystyle \infty $, is a (profinite) normal extension of $\displaystyle \mathbb Q.$
    And show that its Galois group is the additive group of the integer 2-adic number.
    Last edited by ZetaX; Apr 20th 2009 at 08:18 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 $\displaystyle \mathbb Q_{2^n}\cap \mathbb R$, where $\displaystyle n$ is from $\displaystyle 3$ to $\displaystyle \infty $, is a (profinite) normal extension of $\displaystyle \mathbb Q.$
    And show that its Galois group is the additive group of the integer 2-adic number.
    well, $\displaystyle K=\bigcup (\mathbb{Q}_{2^n} \cap \mathbb{R})$ is normal because $\displaystyle \mathbb{Q}_{2^n}/\mathbb{Q}$ is abelian and thus each $\displaystyle K_n=\mathbb{Q}_{2^n} \cap \mathbb{R}$ 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: $\displaystyle \text{Gal}(K/\mathbb{Q}) \cong \varprojlim \text{Gal}(K_n/\mathbb{Q}).$ now the question is what exactly is $\displaystyle \text{Gal}(K_n/\mathbb{Q})$ ?
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