\documentclass{amsart}

\title{Algebra 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 $R$ be a ring. The {\em formal power series ring} $R[[x]]$ consists
    of infinite expressions $a_0  + a_1 x + a_2 x^2 + \ldots$ with the
    natural operations. Prove that if $R$ is Noetherian then $R[[x]]$ is Noetherian.

\item   Let $n \ge 1$ and let $A$ be an $n \times n$ integer matrix. Let
    $G$ be the subgroup of $\mathbb{Z}^n$ generated by the rows of $A$.
    Prove that $G$ has rank $n$ (as a $\mathbb{Z}$-module) iff $A$ is non-singular,
    and that in this case
    the order of the quotient group $\mathbb{Z}^n/G$ is $\vert \det(A) \vert$.

\item  Let $E$ be a field and let $F$ be algebraic over $E$. Prove that $F$ is algebraically
       closed iff every  polynomial  in $E[x]$ splits over $F$. 

\item  Let $E$ be a field of characteristic $p$ and consider the map $\phi:a \mapsto a^p$.
       Show that $\phi$ is an injective HM from $E$ to $E$. Prove that if $E$ is either
       finite or algebraically closed then $\phi$ is an automorphism of $E$. Prove that
       if $\vert E \vert = p^n$ then $\phi$ is an element of order $n$ in the automorphism
       group of $E$.

\item  Let $E$ be the splitting field for $x^4 + 2$ over $\mathbb{Q}$ (that is to say the
       subfield of the complex numbers generated by the roots). Compute
       $[E : \mathbb{Q}]$ and find an element $\alpha$ such that $E = \mathbb{Q}(\alpha)$.
       Repeat the exercise for $x^4 + 3$. 
          
\item  Prove that every algebraically closed field is infinite. 


\item  Find an example of a field which is isomorphic to a proper subfield 
       of itself.



\end{enumerate}



\end{document}
