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\title{Math 332: Homework 8}
%\author{Your name here --- delete the % at the start of the line}
\date{6.IV.15}

\begin{document}
\maketitle

\begin{exc}
For each of the following pairs of integers $a$ and $b$, use the Euclidean algorithm to determine their greatest common divisor $d$ and write $d$ as a linear combination $ax+by$ of $a$ and $b$.
\begin{enumerate}[(a)]
\item $a=20$, $b=13$.
\item $a=69$, $b = 372$.
\item $a = 91442056588823$, $b = 779086434385541$.
\end{enumerate}
\end{exc}

\begin{prob}
Find a generator for the ideal $(85, 1+13i)$ in $\ZZ[i]$, \emph{i.e.}, a greatest common divisor for $85$ and $1+13i$, by the Euclidean algorithm.  Do the same for the ideal $(47-13i,53+56i)$.
\end{prob}

\begin{prob}
Read pp.229-230 of the book on quadratic integer rings.  Let $F = \QQ(\sqrt{D})$ be a quadratic field and let $\mathcal{O}$ be its associated quadratic integer ring.  Let $N$ be its field norm.
\begin{enumerate}[(a)]
\item Suppose $D=-3$.  Prove that $\mathcal{O}$ is a Euclidean domain with respect to $N$.  (You will likely need to show that every element of $F$ differs from an element of $\mathcal{O}$ by an element show norm is at most $1/3<1$.)
\item  Suppose that $D = -163$.  Prove that $\mathcal{O}$ is not a Euclidean domain with respect to any norm.  (Apply a proof similar to the one in the book for the case $D=-19$.)
\end{enumerate}
\end{prob}

\begin{exc}
Prove that the quotient of a PID by a prime ideal is again a PID.
\end{exc}

\begin{prob}
Let $R$ be an integral domain and suppose that every \emph{prime} ideal in $R$ is principal.  Use the following outline to prove that \emph{every} ideal in $R$ is principal, \emph{i.e.}, $R$ is a PID.
\begin{enumerate}[(a)]
\item Assume that the set of ideals of $R$ that are not principal is nonempty and prove that this set has a maximal element under inclusion.  By hypothesis, this ideal is not prime.  [Use Zorn's lemma!]
\item Let $I$ be an ideal which is maximal with respect to being non principal, and let $a,b\in R$ with $ab\in I$ but $a\notin I$ and $b\notin I$.  Let $I_a = (I,a)$ and let $I_b = (I,b)$.  Define $J = \{r\in R\mid rI_a\subseteq I\}$.  Prove that $I_a = (\alpha$ and $J = (\beta)$ are principal ideals in $R$ with $I\subsetneq I_b\subsetneq J$ and $I_aJ = (\alpha\beta)\subseteq I$.
\item If $x\in I$ show that $x=s\alpha$ for some $s\in J$.  Deduce that $I=I_aJ$ is principal, a contradiction, and conclude that $R$ is a PID.
\end{enumerate}
\end{prob}

\begin{prob}
Let $R$ be a commutative ring with $1$ and let $a,b$ be nonzero elements of $R$.  A \emph{least common multiple} of $a$ and $b$ is an element $e$ of $R$ such that
\begin{enumerate}[(i)]
\item $a\divides e$ and $b\divides e$, and
\item if $a\divides e'$ and $b\divides e'$, then $e\divides e'$.
\end{enumerate}
\begin{enumerate}[(a)]
\item Prove that a least common multiple of $a$ and $b$ (if such exists) is a generator for the unique largest PID contained in $(a)\cap (b)$.
\item Deduce that any two nonzero elements in a Euclidean domain have a  least common multiple which is unique up to multiplication by a unit.
\item Prove that in a Euclidean domain the least common multiple of $a$ and $b$ is $\frac{ab}{(a,b)}$, where $(a,b)$ is the greatest common divisor of $a$ and $b$.
\end{enumerate}
\end{prob}

\begin{prob}
Let $a$ and $b$ be nonzero elements of a PID $R$.  Prove that $a$ and $b$ has a least common multiple.  Now assume hat $R$ is additionally a UFD.  Describe the least common multiple of $a$ and $b$ in terms of the prime factorizations of $a$ and $b$.
\end{prob}

\begin{prob}
\begin{enumerate}[(a)]
\item Prove that the quotient ring $\ZZ[i]/(1+i)$ is a field of order $2$.
\item Let $q\in \ZZ$ be a prime with $q\equiv 3\pmod{4}$.  Prove that the quotient ring $\ZZ[i]/(q)$ is a field with $q^2$ elements.
\item Let $p\in \ZZ$ be a prime with $p\equiv 1\pmod{4}$ and write $p=\pi\bar{\pi}$ for some $\pi \in \ZZ[i]$.  Prove that the hypotheses of the Chinese Remainder Theorem are satisfied, and thus
\[
  \ZZ[i]/(p)\cong \ZZ[i]/(\pi)\times \ZZ[i]/(\bar{\pi})
\]
as rings.  Show that the ring $\ZZ[i]/(p)$ has order $p^2$ and that each of $\ZZ[i]/(\pi)$, $\ZZ[i]/(\bar{\pi})$ is a field of order $p$.
\end{enumerate}
\end{prob}

\begin{chal}
An integral domain $R$ in which every ideal generated by two elements is principal is called a \emph{Bezout domain}.
\begin{enumerate}[(a)]
\item Prove that the integral domain $R$ is a Bezout domain if and only if every pair of elements $a,b\in R$ has a greatest common divisor $d$ in $R$ that can be written as an $R$-linear combination of $a$ and $b$.
\item Prove that every finitely generated ideal of a Bezout domain is principal.
\item Prove that $R$ is a PID if and only if $R$ is a UFD that is also a Bezout domain.  (We have proven in class that every PID is a UFD, and PIDs are obviously Bezout domains.  Thus it only remains for you to prove that UFD's which are Bezout domains are in fact PIDs.  Let $0\ne a\in I\normal R$ where $a$ has the minimal number of irreducible factors amongst elements of $I$.  Prove that $I=(a)$ by showing that if there is an element $b\in I$ that is not in $(a)$, then $(a,b)=(d)$ leads to a contradiction.
\end{enumerate}
\end{chal}

\end{document}