\documentclass{amsart}

\title{Field Theory Homework Set II}


\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. 

    






\begin{enumerate}


\item  Let $F$ be a field, and let $Aut(F)$ be the set of automorphisms of
  $F$.

\begin{enumerate}


\item   Show that $Aut(F)$ forms a group under composition.

\item   Show that if $X \subseteq Aut(F)$ and we define
   $Fix(X) = \{ a \in F : \forall \sigma \in X \; \sigma(a) = a \}$ 
   then $Fix(X)$ is a subfield of $F$. 

\item   Show that $Fix(X) = Fix( \langle X \rangle)$, where as usual
      $\langle X \rangle$ is the subgroup generated by $X$. 

\end{enumerate} 
 

\item  Let $E = {\Bbb Z}/2 {\Bbb Z}$. 

\begin{enumerate}

\item  Find an irreducible element in $E[x]$ of degree $3$. 

\item   Construct a finite field with $8$ elements.

\item   Find all the subfields of the field you just constructed, and also
        find its automorphism group.


\end{enumerate} 


\item   What is the minimal polynomial  of $\sqrt{2} + \sqrt{3}$
      over each of the fields
\begin{enumerate}
\item  $\mathbb{Q}$?
\item  $\mathbb{Q}(\sqrt{2})$?
\item $\mathbb{Q}(\sqrt{2}, \sqrt{3})$?
\end{enumerate}


\item    Let $\mathbb{Z}$ be a subring of $S$ and let $a \in S$. $a$ is {\em integral}
  over $\mathbb{Z}$ iff there is $g \in \mathbb{Z}[x]$ such that $g$ is monic and $g(a) = 0$. 

  Show that $a$  is integral over $\mathbb{Z}$  iff  $(\mathbb{Z}[a], +)$  is a finitely
  generated abelian group, where as usual
  $\mathbb{Z}[a] = \{ f(a) : f \in \mathbb{Z}[x] \}$ is the least subring of $S$ containing
  $\mathbb{Z} \cup \{ a \}$.

 
\item   Prove that if $f \in \mathbb{R}[x]$ has  odd degree then $f$ has at
  least one root. Hint: use calculus.


\item    Prove that $\mathbb{Z}[x]$ is not a PID. (Harder) What are the prime
     ideals?

\item   Prove that $\mathbb{C}$ is a vector space over $\mathbb{R}$.  What is
  its dimension? Find a basis.  For each $z \in \mathbb{C}$  prove that
   the map which takes $a$ to $z a$ is linear; also find its trace and
   determinant. 









\end{enumerate}

\end{document}