\documentclass{amsart}

\title{Field Theory Homework Set I}


\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 $R$ be an ID and let $P$ be a prime ideal of $R$. Let $F$ be the
  field of fractions of $R$, and let $S$ be the subset of $F$ consisting of
  elements that can be written in the form $a/b$ where $a \in R$, $b \in R
  \setminus P$.

\begin{enumerate}

\item  Prove that $S$ is a subring of $F$, and that $R$ is contained in $S$
  (as usual, each $r \in R$ is identified with the fraction $r/1$ in $F$).

\item  Prove that the units of $S$ are precisely the elements of form
       $a/b$ where $a, b \in R \setminus P$.

\item  Prove that the set of nonunits in $S$ forms an ideal. 

\item  Prove that the set of nonunits is the only maximal ideal in $S$. 

\item  Suppose now that $R = \mathbb{Z}$, $F = \mathbb{Q}$ and $P = (p)$ for
  some prime number $p$.

 Prove that
\begin{enumerate}
\item $p$ and its associates are the only irreducibles in $S$.
\item The only ideals in $S$ are those of form $(p^n)$ for $n \ge 0$.
\end{enumerate} 

\end{enumerate} 


\item  Let $R = {\Bbb Z}[i]$, the least subring of the complex numbers
  containg $\Bbb Z$ and $i$. 

\begin{enumerate}

\item  Show that $R$ consists of all complex numbers of the form $a + b i$
  with $a, b \in \mathbb{Z}$.

\item  Show that if $a + b i \neq 0$ then the principal ideal $(a + b i)$,
   when considered as a subset of the complex plane, forms
   a square lattice.  Deduce that $R$ is a Euclidean domain (hint:
   use the absolute value as your Euclidean function).  

\item  Let $N(a + b i) = a^2 + b^2$. Show that $N(r s) = N(r) N(s)$ for all
       $r, s \in R$.  
       Show that the units of $R$ are precisely those $r \in R$ with $N(r) =
       1$, and identify them. 
       


\item  Show that $N(r)$ is never congruent to $3$ modulo $4$. use this to show
  that  if the prime number $p$ is congruent to $3$ modulo $4$, then $p$
  is prime in $R$. 

\item   Show that $5$ is not prime in $R$, and find its prime factorisation.  

\end{enumerate}



\item  Let $\alpha = i \sqrt{5}$, and  
 let $R = {\Bbb Z}[\alpha]$, the least subring of the complex numbers
  containg $\Bbb Z$ and $\alpha$. 

\begin{enumerate}

\item  Show that $R$ consists of all complex numbers of the form $a + b \alpha$
  with $a, b \in \mathbb{Z}$.


\item  Let $N(a + b \alpha) = a^2 + 5  b^2$. Show that $N(r s) = N(r) N(s)$. 
       Show that the units of $R$ are precisely those $r \in R$ with $N(r) =
       1$, and identify them. 

\item  Show that $2, 3, 1 + \alpha, 1 - \alpha$ are all irreducible in $R$.

\item  Show that $R$ is not a UFD.  Hint: what is $(1 + \alpha)(1-\alpha)$?

\item  Show that $2, 3, 1 + \alpha, 1 - \alpha$ are not prime in $R$.

\end{enumerate}



\item   Prove that the identity map is the only automorphism of the field
  $\mathbb{R}$.

\item   Let $\mathbb{Q}(i)$ be the least subfield of $\mathbb{C}$ containing
  $\mathbb Q$ and $i$, and $\mathbb{Q}[i]$ be the least subring of $\mathbb{C}$ containing
    $\mathbb Q$ and $i$.  Prove that $\mathbb{Q}(i) = \mathbb{Q}[i]$.
   

\item   Recall that $S_4$ is the group of all permutations of the set $\{1,
  \ldots 4\}$.  Find all the subgroups of $S_4$, and indicate which ones are
   normal.    




\end{enumerate}



\end{document}