# Subgroup

• Jan 9th 2010, 06:46 AM
Sampras
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 $\displaystyle H$?
• Jan 9th 2010, 09:25 AM
NonCommAlg
Quote:

Originally Posted by Sampras
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 $\displaystyle F$?
• Jan 9th 2010, 09:30 AM
Sampras
Quote:

Originally Posted by NonCommAlg
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 $\displaystyle H$.
• Jan 9th 2010, 07:18 PM
NonCommAlg
Quote:

Originally Posted by Sampras
$\displaystyle G(E/F)$ 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. $\displaystyle x \in E_H.$