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\begin{center} {\Large\bf Algebra V63.0344 \\ Assignment 2} \\ Due,
Friday, Feb 7 in Recitation \end{center}
\beq
Many persons who have not studied mathematics confuse it with
arithmetic and consider it a dry and arid science. Actually,
however, this science requires great fantasy. \\ -- Sophia Kovalevsky
\enq
\ben
\item In $Z_7[x]$ let $f(x)=2x^5+3x^4+4x+4$ and $g(x)=3x^2+x+5$. Find
$q(x),r(x)$ with $f(x)=q(x)g(x)+r(x)$ and either $r(x)=0$ or $r(x)$ having
smaller degree than $g(x)$.
\item (*) What is the remainder when $x^{1000000}$ is divided by $x^3+x+1$
in $Z_2[x]$. (There is a pattern!)
\item Further problems on $Z[\omega]$, as in assignment 1.
\ben
\item What are the possible values of $|\ah|^2$, $\ah\in Z[\omega]$,
with $|\ah|^2\leq 11$.
(Notation: For $\ah=a+bi\in C$, $|\ah|=\sqrt{a^2+b^2}$,
the distance from $\ah$ to the origin on the complex plane.)
\item Factor the numbers $2,3,4,5,6,7,8,9,10,11$ into irreducibles in
$Z[\omega]$. Prove that you do indeed have irreducibles. [One aid: If
$\ah=\beta\gamma$ then $|\ah|^2=|\beta|^2|\gamma|^2$. Also, $|\ah|^2=1$
exactly for the six units of assignment 1.]
\item Find the minimal positive real $x$ with the following property:
For {\em any} $\beta\in C$ there exists $\ah=a+b\omega\in Z[\omega]$
with $|\beta-\ah|\leq x$. (The geometry of $Z[\omega]$ is particularly
helpful here.)
\item Use the above and following the argument for $Z[i]$, prove that
$Z[\omega]$ is a Euclidean Ring under $d(\ah)=|\ah|^2$.
\een
\item Even more problems on $Z[\omega]$, as in assignment 1.
\ben
\item In the picture of $Z[\omega]$ (you can use the posted solution
to Assignment 1 or your own picture)
mark (with a little circle) those points which are
in the ideal $(2)$.
\item Describe $Z[\omega]/(2)$ as $\ol{\ah_1},\ldots,\ol{\ah_r}$ for
some specific $\ah_1,\ldots,\ah_r$. (You have to figure out what $r$
is!)
\item Give the multiplication table for $Z[\omega]/(2)$. Is it a
field? (It is automatically a ring so to be a field every element
has to have a multiplicative inverse.)
\een
\item Lets call an integral Domain $D$ together with a function
$d: D-\{0\}\ra \{0,1,2,\ldots\}$ a Banana Domain (not its real
name!) if
\ben
\item $d(\ah)\leq d(\ah\beta)$ for all nonzero $\ah,\beta\in D$
\item If $\ah,\beta\in D-\{0\}$ and if there does not exist $q\in D$
with $\ah=q\beta$ {\em then} there exist $a,b\in D$ with $a\ah+b\beta\neq 0$
and (critically!) $d(a\ah+b\beta)< d(\beta)$.
\een
Prove that a Banana Domain is a P.I.D. . (Hint: Follow
the argument that a Euclidean Domain is a P.I.D.)
\een
\beq Math is natural. Nobody could have invented the mathematical
universe. It was there, waiting to be discovered, and its
crazy; its bizarre.
-- John Conway
\enq
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