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\begin{document}
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\noindent
\bf MATH 307
Final Examination (Take home component).
\rm {\hfill Due: May 8, Monday 5:00 p.m.}
\begin{enumerate}
\medskip
\item Let $R_1$ and $R_2$ be rings, and $\phi: R_1 \rightarrow R_2$ be
a ring homomorphism. {\bf Prove or disprove} the following.
(a) If $A$ is an ideal of $R_1$, then $\phi(A)$ is an ideal in $R_2$.
(b) If $B$ is an ideal of $R_2$, then
$\phi^{-1}(B) = \{a \in R_1: \phi(a) \in B\}$ is an ideal of $R_1$.
\medskip
\item Let $\bD$ be an integral domain with unity 1.
{(a)} Show that $\tilde \bD = \{r\cdot 1: r \in \ZZ\}$ is a subdomain of $\bD$,
\bf and \rm $\tilde \bD$ is contained in every subdomain
\quad of $\bD$. (Make sure that that subdomain
has same unity.)
{(b)} Show that the characteristic of any subdomain of $\bD$
is the same as that of $\bD$.
\medskip
\item Let $\IF$ be a fields. Suppose $f(x) \in \IF[x]$, and
$A = \langle f(x) \rangle = \{f(x) h(x): h(x) \in \IF[x]\}$.
(a) If $f(x) \in \IF[x]$ is reducible\footnote{That is, $f(x) = f_1(x) f_2(x)$
such that $f_1(x), f_2(x)$ has degrees strictly smaller than that of $f(x)$.},
show that the factor ring $\IF[x]/A$ is \bf not \rm an integral domain.
\quad
(It follows that $\IF[x]/A$ is not a field.)
(b) Suppose $f(x)$ is irreducible. Show that
$A$ is an maximal ideal.
\quad (It follows that $\IF[x]/A$ is a field.)
[Hint: If $A$ is not maximal, then there is an ideal $B$ in $\IF[x]$
not equal to $A$ or $\IF[x]$ such that
$A \subset B \subset \IF[x]$. Suppose $g(x) = g_0 + \cdots + g_mx^m$ with
$g_m = 1$ is a polynomial in $B$ with {\bf minimum} degree. Show that
$m> 1$ and $f(x) = g(x) h(x)$ such that $h(x)$ has degree larger than 1 by
the division algorithm Theorem 16.2.]
\medskip
\item Let $A = \langle x^2+1 \rangle \subseteq \ZZ_3[x]$, and ${\bf E} = \ZZ_3[x]/A$.
{(a)} Show that $f(x) = x^2+1$ is irreducible.
[Only need to show that $f(a) \ne 0$ for all $a \in \ZZ_3$.]
{(b)} Determine (with proof) all the generators of
the cyclic group $(\bE^*, \dot)$.
\qquad
[Hint: $\bE^*$ is isomorphic to $\ZZ_8$, and should have
4 generators, each has order 8.]
(c) Find the inverse of $2x+1 + A$ in $\bE$.
\bigskip
\centerline{\bf Good Luck!}
\end{enumerate}
\end{document}