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\title{Algebra Homework Set 8}
\author{James Cummings (jcumming@andrew.cmu.edu)}


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   Due by class time on Wednesday 9 November. Homework must be typeset and submitted by email as a PDF file. 

\begin{enumerate}

\item Let $R$ be a ring and let $I$ and $J$ be ideals such that
  $I + J = R$. Prove that:
\begin{enumerate}
\item  $I J = I \cap J$, where $I J$ is the ideal consisting of all elements of form $\sum_{i=1}^n a_i b_i$ with
    $a_i \in I$, $b_i \in J$. 
\item  $R/I J \simeq R/I \oplus R/J$ as an $R$-module.
\item  $R/I J \simeq R/I \oplus R/J$ as a ring.  
\end{enumerate} 

\item Prove that if $k$ is a field, $f \in k[x]$ is a polynomial with $deg(f) > 0$ and
the irreducible factorisation of $f$ is $g_1 \ldots g_n$ then:
\begin{enumerate}
\item $k[x]/(f)$ is isomorphic to $\bigoplus_{i = 1}^n k[x]/(g_i)$ as a $k[x]$-module.
\item $k[x]/(f)$ is isomorphic to $\displaystyle\prod_{i = 1}^n k[x]/(g_i)$ as a ring.
\end{enumerate}   

\item Let $k$ be a field, let $\lambda \in k$, let $f = (x - \lambda)^n$ for some $n > 0$ and consider
  the quotient $V = k[x]/(f)$ as a $k[x]$ module. Show that $\{ (x - \lambda)^i + (f) : 0 \le i < n \}$ forms a basis for
  $V$ as a $k$-vector space, and find the matrix of the $k$-linear map from $V$ to $V$
  given by $v \mapsto x v$.

\item Prove that if $k$ is a field then:
\begin{enumerate} 
\item  The polynomial ring $k[x]$ contains infinitely many pairwise non-associate irreducible elements. Hint: Euclid.
\item  There do not exist elements $a_1, \ldots a_n \in k(x)$ such that
   $k(x) = k[a_1, \ldots a_n]$.
\end{enumerate}

\item  Prove that if $l$ is an algebraic extension field of $k$ with $a, b \in l$ 
  and the numbers $[k(a): k]$ and $[k(b): k]$ are coprime then
  $[k(a, b): k] = [k(a): k] \times [k(b) : k]$.  


  
\end{enumerate}
  
  
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