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\title{MATH0037 Galois Theory 2002, Sheet 2}
\author{Geoff Smith, http://www.bath.ac.uk/$\sim$masgcs}
\date{}
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\maketitle
\begin{enumerate}

\item Let $q$ be a rational number. 
Define a map $\varepsilon_q : \mathbb Q [X ] \rightarrow \mathbb Q$ via
$f \mapsto f(q)$. Show that $\varepsilon_q$ is a ring homomorphism.
Describe $\mbox{Ker } \varepsilon_0$ explicitly.
\item Let $\mathbb Z_2$ denote the integers modulo 2.
In lectures we constructed \[\mathbb R[X]/(X^2+1)\] which turned out
to be $\mathbb C$ in disguise. Construct the rings
\begin{enumerate}
\item $\mathbb Z_2 [X]/(X^2+1)$ and
\item $\mathbb Z_2 [X]/(X^2 + X + 1).$
\end{enumerate}
Each of these structures is a ring with 4 elements. One of 
them is a field and the other is not. Which is which, and further
more, what properties of the polynomials $X^2+1$ and $X^2 + X + 1$
(in $\mathbb Z_2[X]$) are causing this difference?
\item Construct $\mathbb R[X]/(X^2 + X + 1).$ Can you recognize
this structure?
\item Construct $\mathbb R[X]/(X^2 - 3X + 2).$ Can you recognize
this structure?
\item Construct addition and multiplication tables for a field
which has exactly 9 elements.
\item In this question only, $R$ not guaranteed to have
a commutative law of multiplication, but all the other
ring axioms are satisfied.
\begin{enumerate}
\item[(a)] Suppose that $r^2 = r\ \forall r \in R$. Prove that
$R$ has commutative multiplication.
\item[(b)$^*$] Suppose that $r^3 = r\ \forall r \in R$. Prove that
$R$ has commutative multiplication.
\end{enumerate}
\end{enumerate}
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