\documentclass{amsart}

\title{Field Theory Homework Set III}


\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  Find a basis for $F = \mathbb{Q}(i, \sqrt{2})$ over $\mathbb{Q}$. 
        Compute the group $Aut(F/\mathbb{Q})$. 
        Find all its subgroups, and describe their fixed fields.

 
\item  Prove that if $F$ is a finite field it has size $p^n$ where
    $p$ is the characteristic and $n > 0$.  


\item  Show that a polynomial $f(x)$ is irreducible in $\mathbb{Z}[x]$  iff $f(x+1)$ is irreducible.
    Use this to show that  $(x^p - 1)/(x-1)$ is irreducible
    for every prime $p$.

\item   Find an algebraic extension of $\mathbb{Q}$ which is not of finite degree.

\item Let $p$ be an odd prime. Prove that $-1$ has a square root mod $p$ iff $p \equiv 1 \mod 4$. 

\item (Challenging) Prove that if $p$ is an odd prime and $p \equiv 1 \mod 4$ then $p$ is
  the sum of two perfect squares.  Hint: use some old HW about the ring
  $\mathbb{Z}[i]$, think about what happens to primes of $\mathbb{Z}$ in this
  bigger ring. 




\end{enumerate}

\end{document}