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\title{Math 721\\Spring 2011\\Homework 5}
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\centerline{\it Due Monday April 4, 2011.}

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\noindent 1. (5 pt) Let $K\subseteq F$ be fields. We define $\text{Aut}_K(F)$ to be the set of automophisms \\ $\sigma:F\longrightarrow F$ such that $\sigma(k)=k$ for all $k\in K$. This is called the {\it Galois group} of $F$ over $K$. Show that if $[F:K]=n$ then $\vert\text{Aut}_K(F)\vert$ divides $n$. What happens when $n$ is prime?

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\noindent 2. (5 pt) Show that if $K\subseteq F$ and $u\in F$ is algebraic over $K$ of odd degree, then $K(u^2)=K(u)$. What can you say if $u$ is transcendental over $K$?

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\noindent 3. (5 pt) Let $F$ be a finite field of characteristic $p$. Show that $\text{Aut}_{\mathbb{Z}_p}(F)$ is cyclic.

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\noindent 4. (5 pt) Let $K\subseteq F$ be fields and let $\overline{K}_F=\{z\in F\vert z \text{ is algebraic over K}\}$. Show that $\overline{K}_F$ is a subfield of $F$ containing $K$ (this is called the algebraic closure of $K$ in $F$).




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