\documentclass{amsart}

\title{Fields Homework Set V}


\author{James Cummings}



\begin{document}


\maketitle

    You may collaborate on this homework set, but must write up your solutions
    by yourself. Please contact me by email if you are puzzled by something,
    would like a hint or believe that you have found a typo. 

    Due March 31st. 





\begin{enumerate}


\item Let $F$ be an extension of $E$, let $\rho \in Aut(F/E)$ and $H \le Aut(F/E)$.
      What is the relationship between $Fix(H)$ and $Fix(H^\rho)$?

\item   Complete the calculation (mostly done in class) of all the intermediate fields
     for the splitting field of $x^5 -2$ over $\mathbb{Q}$.

\item  Let $E$ be the subfield of the complex numbers generated over $\mathbb{Q}$ by the
   roots of $x^4 + 2$. Compute the degree $[E :\mathbb{Q}]$. Find all the intermediate fields
   of this extension. Which ones are Galois extensions of $\mathbb{Q}$?

\item  Let $p$ be prime, let  $E = \mathbb{Z}/p \mathbb{Z}$ and let
       $f = x^{p^n}- x$ for some $n > 0$.
       Prove that  $f$ has no repeated roots in any extension of $E$. Prove that
       if $F$ is a splitting field for $f$ over $E$ then the roots of $f$ form a 
       subfield of $F$ and use this to conclude that $\vert F \vert = p^n$.

\item  Let $F$ be an algebraic extension of $E$. Prove that the set of $a \in F$
       which are separable over $E$ forms a subfield of $F$. 



\end{enumerate}



\end{document}
