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\name{}
\class{Math 171 - Abstract Algebra I Section \uline{\;\;}}
\assignment{HW \# 16}
\duedate{11/4/10}

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\begin{document}

\problemlist{Dummit \& Foote (8.3) 1, 2, 8, 11}

\begin{problem}[8.3.1]
Let $G = \Q^\times$ be the multiplicative group of nonzero rational numbers.  If $\alpha = p/q \in G$, where $p$ and $q$ are relatively prime integers, let $\varphi: G \to G$ be the map which interchanges the primes 2 and 3 in the prime power factorizations of $p$ and $q$ (so, for example, $\varphi(2^43^{11}5^113^2) = 3^42^{11}5^113^2$, $\varphi(3/16) = \varphi(3/2^4) = 2/3^4 = 2/81$, and $\varphi$ is the identity on all rational numbers with numerators and denominators relatively prime to 2 and to 3).
\begin{itemize}
\item[(a)] Prove that $\varphi$ is a group isomorphism.
\item[(b)] Prove that there are infinitely many isomorphisms of the group $G$ to itself.
\item[(c)] Prove that none of the isomorphisms above can be extended to an isomorphism of the {\it ring} $\Q$ to itself.  In fact prove that the identity map is the only ring isomorphism of $\Q$.
\end{itemize}
\end{problem}
\newpage

\begin{problem}[8.3.2]
Let $a$ and $b$ be elements of the Unique Factorization Domain $R$.  Prove that $a$ and $b$ have a least common multiple (cf. Exercise 11 of Section 1) and describe it in terms of the prime factorizations of $a$ and $b$ in the same fashion that Proposition 13 describes their greatest common divisor.
\end{problem}
\newpage

\begin{problem}[8.3.8]
Let $R$ be the quadratic integer ring $\Z[\sqrt{-5}]$ and define the ideals $I_2 = (2, 1+ \sqrt{-5})$, $I_3 = (3, 2 + \sqrt{-5})$, and $I_3' = (3, 2 - \sqrt{-5})$.
\begin{itemize}
\item[(a)] Prove that $2,\ 3,\ 1 + \sqrt{-5}$ and $1 - \sqrt{-5}$ are irreducibles in $R$, no two of which are associate in $R$, and that $6 = 2 \cdot 3 = (1 + \sqrt{-5}) \cdot (1 - \sqrt{-5})$ are two distinct factorizations of 6 into irreducibles in $R$.
\item[(b)] Prove that $I_2$, $I_3$, and $I_3'$ are prime ideals in $R$.  [One approach: for $I_3$, observe that $R/I_3 \cong (R/(3))/(I_3/(3))$ by the Third Isomorphism Theorem for Rings.  Show that $R/(3)$ has 9 elements, $(I_3/(3))$ has 3 elements, and that $R/I_3 \cong \Z/3\Z$ as an additive abelian group.  Conclude that $I_3$ is a maximal (hence prime) ideal and that $R/I_3 \cong \Z/3\Z$ as rings.]
\item[(c)] Show that the factorizations in (a) imply the equality of ideals $(6) = (2)(3)$ and $(6) = (1 + \sqrt{-5})(1 - \sqrt{-5})$.  Show that these two ideal factorizations give the same factorization of the ideal $(6)$ as the product of prime ideals (cf. Exercise 5 in Section 2).
\end{itemize}
\end{problem}
\newpage

\begin{problem}[8.3.11]
({\it Characterization of P.I.D.s}) Prove that $R$ is a P.I.D. if and only if $R$ is a U.F.D. that is also a Bezout Domain (cf. Exercise 7 in Section 2).  [One direction is given by Theorem 14.  For the converse, let $a$ be a nonzero element of the ideal $I$ with a minimal number of irreducible factors.  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{problem}

\end{document}