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\begin{center} {\Large\bf Honors Algebra , Assignment 10} \\
Due Friday, November 13
\end{center}
\begin{quote}
The truth may be puzzling. It may take some work to grapple with. It
may be counterintuitive. It may contradict deeply held prejudices. It
may not be consonant with what we desperately want to be true. But our
preferences do not determine what's true. We have a method, and that
method helps us to reach not absolute truth, only asymptotic approaches
to the truth; never there, just closer and closer, always finding vast
new oceans of undiscovered possibilities.
\\ -- Carl Sagan
\end{quote}
\ben
\item Let $R$ be a ring and $M\subset R$ an ideal. Assume $M\neq R$
(but do {\em not} assume $M$ is maximal). Let $a\in R$ with $a\not\in M$.
\ben
\item
{\em Assume} there exists an ideal $N$ with $M\subset N \subset R$ and
$N\neq M,R$ and $a\in N$.
Prove that $\ol{a}$ has no multiplicative inverse in $R/M$.
\item {\em Assume} there does {\em not} exist an ideal $N$ with $M\subset N \subset R$ and
$N\neq M,R$ and $a\in N$. Prove that $\ol{a}$ has an multiplicative inverse in $R/M$
\een
\item Let $R$ be a ring and let $a,b\in R$. Set
\[ M = \{ar+bs: r,s\in R\} \]
Prove that $M$ is an ideal. {\tt Notation:} We will write $M=(a,b)$.
\item Let $Z[i]=\{a+bi: a,b\in Z\}$ where $i=\sqrt{-1}$, the usual
{\em Gaussian Integers}.
\ben
\item For $\ah\in Z[i]$ define (this is called a {\em norm})
$N(\ah)=|\ah|^2$, where $|\cdot|$ is the
usual complex number absolute value, that is
$|c+di|=\sqrt{c^2+ d^2}$.
Give a formula for $N(\ah)$ for $\ah\in Z[i]$.
Show $N(\ah\beta)= N(\ah)N(\beta)$.
\item Precisely which elements of $Z[i]$ have multiplicative inverses?
(Use the norm to show that you have everything.)
\item Define $\phi:Z[i]\ra Z[i]$ by $\phi(a+bi)=a-bi$. (This is
generally known as {\em complex conjugation}.) Show that $\phi$ is
a homomorphism by showing $\phi(\ah+\beta)=\phi(\ah)+\phi(\beta)$
and $\phi(\ah\beta)=\phi(\ah)\phi(\beta)$. Show that $\phi$
has kernel $\{0\}$.
\item A number $\ah\in Z[i]$ is called {\em composite} if we can
write $\ah=\beta\gamma$ where neither $\beta$ not $\gamma$ have
multiplicative inverses. (That last condition is to avoid
``trivial" factorizations like $23 = i(-23i)$.) If it is nonzero,
not a unit, and not composite it is called {\em prime}.
Show that
$2$ is composite. Show that $41$ is composite.
Show $7+2i$ is prime. (Idea: Use the norm)
\een
\item Set
\[ R = \{ a + b\sqrt{-5}: a,b\in Z \} \]
\ben
\item
Give a formula for $N(\ah)$ (as defined above) for $\ah=a+b\sqrt{-5}\in R$.
\item Precisely which elements of $R$ have multiplicative inverses?
(Use the norm to argue that you have all of them.)
\item Set \[ I = \{2\ah+(1+\sqrt{-5})\beta: \ah,\beta\in R\} \]
Plot those $(a,b)$ with $-4\leq a,b\leq +4$ so that
$a+b\sqrt{-5}\in I$.
\item Show that I is an ideal.
\item Show that $1\not\in I$.
\item Show that $I$ is {\em not} a principal ideal. (Idea: Assume
$I=(\kappa)$ and use the properties of $N(\cdot)$ above.)
\item Find representatives of $R/I$. What well known field is it
isomorphic to?
\een
\een
\begin{quote}
My father took his stick and began writing an equation in the sand.
Like all the rest of them, this one was busy with $x$'s and $y$'s
resting on top of one another on dash-shaped bunks. Letters were
multiplied by symbols, crowded into parentheses, and set upon by
dwarfish numbers drawn at odd angles.
\\ David Sedaris, Genetic Engineering
\end{quote}
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