
\documentclass{article}
\pagestyle{empty}
\usepackage{amssymb}
\usepackage{latexsym}

\title{MATH0037 Galois Theory 2000}
\author{GCS -- Sheet 2}
\date{}
\begin{document}
\maketitle

{\it All rings are
commutative with 1 and $0$ is not a natural number as 
far as this course is concerned}
\begin{enumerate}
\item Consider $f, g \in \mathbb Q[X]$ where $f = x^4 + 4x +1$
and $g = x^2 +x -2$. Find $q, r \in \mathbb Q[X]$
such that $f = qg + r$ and $\hbox{deg }r < \hbox{deg }g$.
\item Find $f \in \mathbb Q[X]$ of smallest possible degree
with the property that when divided by $X-j$ (in $\mathbb Q[X]$)
it leaves remainder $j$ for each $j \in \{1,2,3,4\}$.
\item Suppose that $f \in \mathbb C[Y]$ is a polynomial
with the property that $\mathbb C[Y]/(f)$ and  
$\mathbb C[Y]/(f^2)$ are isomorphic rings. What can you
deduce about $f$?
\item Suppose that $R$ is a ring and that $I_1, I_2$ are ideals
of $R$. For $i=1,2$ we let $\pi_i$ denote the
natural epimorphism $\pi_i : R \rightarrow R/I_i$ where
$(r)\pi_i = I_i + r$ for every $r \in R$.
Define $\pi : R \rightarrow R/I_1 \oplus R/I_2$ by
$(r)\pi = ( (r)\pi_1, (r)\pi_2)$ for every $r \in R.$
\begin{enumerate}
\item Prove that $\pi$ is a ring homomorphism.
\item Prove that $\pi$ is injective if and only if
$I_1 \cap I_2 = \{ 0\}$.
\item Prove that $\pi$ is surjective if and only if
$I_1 + I_2 = R$.
\item Prove that the rings $\mathbb Z/(6)$
and $\mathbb Z/(2) \oplus \mathbb Z/(3)$ are isomorphic.
\end{enumerate} 
\item Suppose that $R$ is a ring and that $I, J$ are ideals
of $R$ with the property that $I + J = R$. Suppose that
$n, m$ are natural numbers.
Prove that $I^m + J^n = R$. {\it Hint $1^{m+n} = 1$}.
\end{enumerate}
\end{document}

