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\title{MATH0037 Galois Theory 2000}
\author{GCS -- Sheet 1}
\date{}
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\maketitle

{\it All rings are
commutative with 1 as far as this course is concerned}
\begin{enumerate}
\item Suppose that $R$ is a ring and that $I$ and $J$ are both ideals
of $R$. Let $I + J$ be the subset
$\{ a + b \mid a \in I,\ b \in J \}$ of
$R$. Prove that $I + J \unlhd R.$
\item Suppose that $R$ is a ring and
that $I$ and $J$ are both ideals
of $R$. Let $IJ$ be the set of all sums
\[\sum_{i=1}^m a_i b_i\]
where $m$ is any natural number, and for each $i$ we have
$a_i \in I$ and $b_i \in J$. Prove that $IJ \unlhd R.$

\item Suppose that $R$ is a ring and that $I$ is an ideal
of $R$. The previous question given meaning to the ideal
$II$ and we refer to this ideal as $I^2$. How should we define
$I^3$? Give a sensible definition
of an ideal $I^n$ when $n$ is any natural number.

\item Recall that every subgroup of a cyclic group is cyclic.
Use this result to classify (list and understand)
the additive subgroups of $\mathbb Z$
(the additive group of integers). Now classify the
ideals of the ring $\mathbb Z$.

\item Suppose that $m$ is an integer. Let
$(m)$ denote the set of integer multiples of
$m$, so $(m) = \{km \mid k \in {\mathbb Z}\}$.
\begin{enumerate}
\item How many elements are there in the ring ${\mathbb Z}/(2)$?
\item How many elements are there in the ring ${\mathbb Z}/(m)$?
\item Write out the addition and multiplication tables of
${\mathbb Z}/(3)$ (it may be helpful to write
${\bold 0} := (3) + 0,$
${\bold 1} := (3) + 1$ and
${\bold 2} := (3) + 2$.
\end{enumerate}

\item Suppose that $R$ is a ring and that $I$ and $J$ are both ideals
of $R$. Show that $I \cup J$ is an ideal of $R$ if and only if
$I \cap J \in \{ I, J\}$.

\item An ideal $I$ of a ring $R$ is called a
{\it principal ideal} if there is $r \in R$ such that
$I = (r)$ where $(r) = \{sr \mid s \in R\}$.
Find an ideal of a ring which is not a principal ideal
of that ring. {\it Hint: Look in ${\mathbb Z}[X]$, the
ring of polynomials in the variable $X$ with
integral coefficients.}

\item Let ${\mathbb C}[[X]]$ denote the set of formal power series
in $X$ with coefficients in ${\mathbb C}.$ These series are simply
expressions of the form
\[ a_0 + a_1 X + a_2 X^2 + \ldots \]
where we pay no heed to convergence questions. Show how to
define addition and multiplication on ${\mathbb C}[[X]]$
in a natural way so that it becomes a ring.

When you have done this, show that $f = f_0 + f_1 X + \ldots \in
{\mathbb C}[[X]]$ has a multiplicative inverse
in ${\mathbb C}[[X]]$ if and only
if $f_0 \not = 0$.

\end{enumerate}
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