\documentclass{amsart}

\title{Field Theory Homework Set IV}


\author{James Cummings}


\usepackage{amssymb} 

\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. 


\begin{enumerate}

\item     
 An {\em ordered field} is a field $F$ together with a set $P \subseteq F$ of elements 
   such that (defining $-P = \{ -a : a \in P \}$)

\begin{enumerate}

\item   $P$ is closed under $+$ and $\times$ 
\item   $P \cap  -P = \{ 0 \}$.
\item   $P \cup  -P = F$.

\end{enumerate}

   Intuitively, you can think of $P$ as the set of
   field elements which have been designated as non-negative.

 Prove that 

\begin{enumerate}

\item   There is a unique set $P \subseteq \mathbb{R}$ such that $(\mathbb{R}, P)$ 
  is an ordered field.

\item   There is no set $P \subseteq \mathbb{C}$ such that $(\mathbb{C}, P)$ 
  is an ordered field.

\item   There are at least two sets  $P \subseteq \mathbb{Q}(\sqrt{2})$ such
  that $(\mathbb{Q}(\sqrt{2}), P)$ is an ordered field. 

\item   If $(F, P)$ is an ordered  field then   $F$ has characteristic zero
   and $-1$ is not a sum of squares in $F$.

\end{enumerate}
  

\item  Let $F$ be a field extending the field $E$ and let $a \in F$ be algebraic 
   over $E$. Show that the following are equivalent:
\begin{enumerate}
\item  $a$ is separable over $E$.
\item  There is a field $F'$ extending $E$ such that there exist $[E(a): E]$
      distinct monomorphisms $\sigma: E(a) \rightarrow F'$ with $\sigma \restriction E = id$.
\end{enumerate}

\item  Let $f = x^4 - 2$.

\begin{enumerate}

\item Show that $f$  is irreducible in $\mathbb{Q}[x]$.

\item Find the complex roots of $f$, and show that $f$ splits over the complex numbers. 

\item Let $E$ be the subfield of $\mathbb{C}$ generated by the roots of $f$ (so that
      $E$ is a splitting field for $\mathbb{Q}$).   
      Find $[E : \mathbb{Q}]$.

\item  Compute the group $Aut(E/\mathbb{Q})$, and describe how each element 
     of this group permutes the roots of $f$.  

\item  Find all the subgroups of $Aut(E/\mathbb{Q})$ and their fixed fields.

  
\end{enumerate} 


\item   Let $F$ have characteristic $p$.  Use the binomial theorem to show that
        the map $a \mapsto a^p$ is a monomorphism from $F$ to $F$. Show that if $F$
        is finite this map is an automorphism of $F$, and in that case describe its
        fixed field. 









\end{enumerate}

\end{document}
