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\begin{document}

\noindent{\bf MATHEMATICS} {\bf 361} \hspace{2in} {\bf Name}
\hrulefill

\noindent{\bf Final Examination}

\noindent December 14, 2000 \hfill 6 questions

\next
\parindent=0mm

1. ({\sl 30 points}) Suppose $S, T$ are sets, $\sim$ is an
equivalence relation on $S$, and $f:T \rightarrow S$
is a mapping.

\vspace{3mm}
\noindent (a) Define what it means to say that $\sim$ is an equivalence
relation
on $S$.

\vspace{1.5in}

 (b) Define the equivalence class $[a]$ of an element $a$ of $S$.

\vspace{1in}

 (c) If $x,y \in T$, define $x \approx y$ to mean that $f(x)
\sim f(y)$. Show that $\approx$ is an equivalence relation on
$T$.

\vspace{2in}

 (d) If $C$ is an equivalence class in $T$, show that
$f(C)=\{f(x)|x \in C\}$ is a subset of a single equivalence
class $C'$ in $S$.


\newpage

2. ({\sl 30 points}) Let $m, n$ be integers (not both $0$), and
let
$x$ be an element of a group $G$.

\vspace{2mm}

(a) Define carefully what it means for $m$ and $n$ to be
relatively prime.

\vspace{0.7in}

 (b)  It is true that the following statements are equivalent.

\begin{quotation}(i) $m$ and $n$ are relatively prime.

(ii) There exist integers $a, b$ such that $am+bn=1$.
\end{quotation}
Give a proof that (ii) $\Rightarrow$ (i).

\vspace{2in}
(c) Explain what is meant by the cyclic subgroup of $G$
generated by $x$.

\vspace{1in}
 (d) If $x$ has order $n$, show that $x$ lies in the
subgroup of $G$ generated by $x^m$, if and only if, $m$ and
$n$ are relatively prime.


\vspace{2.7in}

\newpage

3. ({\sl 20 points}) Which of the following is a subgroup of the
additive group of the real numbers? [{\sl No justification
needed.}]
\vspace{4mm}

(a) The set of all positive real numbers.

\vspace{1.2in}

(b) The set of all rational numbers.

\vspace{1.2in}

(c) The set of all irrational numbers.

\vspace{1.2in}

(d) The closed interval [-1,1].

\vspace{1.2in}

(e) The set of all numbers of the form $a+b \pi +c \pi ^2$,
where $a,b,c$ are integers.




\newpage

4. ({\sl 30 points}) Let $G,G'$ be groups, $\theta :G
\rightarrow G'$ a homomorphism, and $K$ the kernel of $\theta$.
\vspace{2mm}

(a) Define what it means to say that $\theta$ is a homomorphism.

\vspace{1in}

(b) Define what it means to say that $K$ is the kernel of
$\theta$.

\vspace{1in}

(c) Show that $K$ is a normal subgroup of $G$.

\vspace{2.7in}

(d) If $a,b \in G, ab^{-1} \in K$, and $n$ is an integer, show
that $a^{n}b^{-n} \in K$.


\newpage

5. ({\sl 20 points}) Let $R$ be a ring (with unit).
\vspace{2mm}

(a) Define what it means for a subset $I$ of $R$ to be a
 left ideal
in $R$.
\vspace{2in}

(b) Suppose $A$ is a subset of $R$ with the property that $$r
\in R, a \in A \Rightarrow ra \in A.$$
Let $I$ be the set of all finite sums $a_1 +a_2+...+a_m$, where
all $a_i \in A$. Prove that $I$ is a
 left ideal of $R$.

\newpage
6. ({\sl 20 points}) Let $F[x]$ be the polynomial ring in $x$
over a field $F$,
$p(x) \in F[x]$, deg $p(x) > 0.$ Let $I=(p(x))$ be the principal
ideal of $F[x]$ generated by $p(x).$\vspace{4mm}

(a) Define what it means to say that $p(x)$ is irreducible over
$F.$\vspace{1.6in}

(b) Define what it means to say that $I$ is a maximal ideal of
$F[x].$\vspace{1.6in}

(c) State two theorems of the course which can be used to show
that $p(x)$ is irreducible over $F$ if, and only if, $F[x]/I$ is
a field.


\end{document}


