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Thread: Subgroup

  1. #1
    Senior Member Sampras's Avatar
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    Subgroup

    Let $\displaystyle E $ be an algebraic extension of a field $\displaystyle F $. Let $\displaystyle S = \{\sigma_{i}: i \in I \} $ be a collection of automorphisms of $\displaystyle E $ such that every $\displaystyle \sigma_i $ leaves each element of $\displaystyle F $ fixed. Show that if $\displaystyle S $ generates a subgroup $\displaystyle H $ of $\displaystyle G(E/F) $, then $\displaystyle E_S = E_H $.

    The group of automorphisms of a field is cyclic and has the Frobenius automorphism as a natural generator. So $\displaystyle H $ is cyclic. Thus the elements left fixed by $\displaystyle S $ are the same as the elements left fixed by $\displaystyle H $?
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  2. #2
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    Quote Originally Posted by Sampras View Post
    Let $\displaystyle E $ be an algebraic extension of a field $\displaystyle F $. Let $\displaystyle S = \{\sigma_{i}: i \in I \} $ be a collection of automorphisms of $\displaystyle E $ such that every $\displaystyle \sigma_i $ leaves each element of $\displaystyle F $ fixed. Show that if $\displaystyle S $ generates a subgroup $\displaystyle H $ of $\displaystyle G(E/F) $, then $\displaystyle E_S = E_H $.

    The group of automorphisms of a field is cyclic and has the Frobenius automorphism as a natural generator. So $\displaystyle H $ is cyclic. Thus the elements left fixed by $\displaystyle S $ are the same as the elements left fixed by $\displaystyle H $?
    what are $\displaystyle E_S, E_H, G(E/F)$ ? i've seen your posts. most of them suffer from the lack of clarity! use standard notation or define clearly what you mean by them. for example, did you by $\displaystyle G(E/F)$

    mean $\displaystyle \text{Gal}(E/F),$ the Galois group of $\displaystyle E$ over $\displaystyle F$?
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  3. #3
    Senior Member Sampras's Avatar
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    Quote Originally Posted by NonCommAlg View Post
    what are $\displaystyle E_S, E_H, G(E/F)$ ? i've seen your posts. most of them suffer from the lack of clarity! use standard notation or define clearly what you mean by them. for example, did you by $\displaystyle G(E/F)$

    mean $\displaystyle \text{Gal}(E/F),$ the Galois group of $\displaystyle E$ over $\displaystyle F$?
    $\displaystyle G(E/F) $ is the group of automorphisms of $\displaystyle E $ that leave $\displaystyle F $ fixed. $\displaystyle E_S $ and $\displaystyle E_H $ are the elements of $\displaystyle E $ left fixed by $\displaystyle S $ and $\displaystyle H $.
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    Quote Originally Posted by Sampras View Post
    $\displaystyle G(E/F) $ is the group of automorphisms of $\displaystyle E $ that leave $\displaystyle F $ fixed. $\displaystyle E_S $ and $\displaystyle E_H $ are the elements of $\displaystyle E $ left fixed by $\displaystyle S $ and $\displaystyle H $.
    well, trivially $\displaystyle E_H \subseteq E_S$ because $\displaystyle S \subseteq H.$ conversely, let $\displaystyle x \in E_S$ and $\displaystyle \sigma \in H.$ then $\displaystyle \sigma = \sigma_{i_1} \sigma_{i_2} \cdots \sigma_{i_n},$ for some $\displaystyle i_1, i_2, \cdots , i_n \in I,$ because $\displaystyle H=<S>.$ now since $\displaystyle x \in E_S,$ we have $\displaystyle \sigma_{i_k}(x)=x,$

    for all $\displaystyle i_k.$ thus $\displaystyle \sigma(x)=x,$ i.e. $\displaystyle x \in E_H.$
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