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\begin{center}
{\bf \Large \underline{Honors Algebra 4, MATH 371 Winter 2010}}\\ 
	{\large Assignment 3}\\{Due Friday, February 5 at 08:35}
\end{center}

\begin{enumerate}
	\item Let $R\neq 0$ be a commutative ring with 1 and let $S\subseteq R$ be the subset of nonzero elements which 
	are not zero divisors. 
	\begin{enumerate}
		\item Show that $S$ is multiplicatively closed. 
		
		\item By definition, {\em the total ring of fractions of $R$} is the ring $\Frac(R):=S^{-1}R$; 
		it is a ring 	equipped with a canonical ring homomorphism $R\rightarrow S^{-1}R$.  
		If $T$ is any multiplicatively closed subset of $R$ that is contained in $S$, show that there 
		is a canonical injective 
		ring homomorphism $T^{-1}R\rightarrow \Frac(R)$, and conclude that $T^{-1}R$ is isomorphic to a subring
		of $\Frac(R)$.  
		
		\item If $R$ is a domain, prove that $\Frac(R)$ is a field and hence that $T^{-1}R$ is a domain 
		for any $T$ as above.
\end{enumerate}

\item Let $R$ be a commutative ring with $1$.
	\begin{enumerate}
		\item Let $S\subseteq R$ be a multiplicatively closed subset.
 		Prove that the prime ideals of $S^{-1}R$ are in bijective correspondence with the
		prime ideals of $R$ whose intersection with $S$ is empty.  
		
		\item If $\mathfrak{p}$ is an ideal of $R$, show that $S:=R\setminus \mathfrak{p}$
		is a multiplicatively closed subset if and only if $\mathfrak{p}$ is a prime ideal.
		Writing $R_{\mathfrak{p}}$ for the ring of fractions $S^{-1}R$, show that 
		$R_{\mathfrak{p}}$ has a unique maximal ideal, and that this ideal is the image of
		$\mathfrak{p}$ under the canonical ring homomorphism $R\rightarrow R_{\mathfrak{p}}$.
		(In other words, the {\em localization of $R$ at $\mathfrak{p}$} is a {\em local ring}).

		\item Let $r\in R$ be arbitrary.  Show that the following are equivalent: 
			\begin{enumerate}
				\item $r=0$
				\item The image of $r$ in $R_{\mathfrak{p}}$ is zero for all
				prime ideals $\mathfrak{p}$ of $R$.
				\item The image of $r$ in $R_{\mathfrak{p}}$ is zero for all
				maximal ideals $\mathfrak{p}$ of $R$.
			\end{enumerate}
	\end{enumerate}
	
	
	\item Do exercises 8--11 in \S7.6 of Dummit and Foote (inductive and projective limits).
	
	\item A {\em B\'ezout domain} is an integral domain in which every finitely generated ideal
	is principal.  
	\begin{enumerate}
		\item Show that a B\'ezout domain is a PID if and only if it is noetherian.
		
		\item Let $R$ be an integral domain.  Prove that $R$ is a Bezout domain if and only if
		every pair of elements $a,b\in R$ has a GCD $d\in R$ that can be written as an $R$-linear combination
		of $a$ and $b$, {\em i.e.} such that there exist $x,y\in R$ with $d=ax+by$.
		
		\item Prove that a ring $R$ is a PID if and only if it is a B\'ezout domain that is also a UFD.
		
		\item Let $R$ be the quotient ring of the polynomial ring $\Q[x_0,x_1,\ldots]$
		over $\Q$ in countably many variables by the ideal $I$ generated by the set 
		$\{x_i-x_{i+1}^2\}_{i\ge 0}$.  Show that $R$
		is a B\'ezout domain which is not a PID (Hint: have a look at Dummit and Foote, \S9.2 \# 12).
		
		{\bf Remark:} The above example of a B\'ezout domain which is not a PID is somewhat artificial. 
		More natural examples include the ``ring of algebraic integers" ({\em i.e.} the set of all
		roots of monic irreducible polynomials in one variable over $\Z$) and the ring of holomorphic functions
		on the complex plane.  The proofs that these are B'ezout domains is, as far as I know, 
		difficult.  For example, in the case of the algebraic integers, one needs the theory of
		class groups).
\end{enumerate}

	\item Let $R=\Z[i]:=\Z[X]/(X^2+1)$ be the ring of {\em Gaussian integers}.
	\begin{enumerate}
		\item Let $N:R\rightarrow \Z_{\ge 0}$ be the {\em field norm}, that is
		$$N(a+bi) := (a+bi)(a-bi)=a^2+b^2.$$  
		Prove that $R$ is a Euclidean domain with this norm.  Hint: there is a proof in the book on pg. 272,
		but you should try to find a different proof by thinking {\em geometrically}.
		
		\item Show that $N$ is multiplicative, {\em i.e.} $N(xy)=N(x)N(y)$ and deduce that $u\in R$
		is a unit if and only if $N(u)=1$.  Conclude that $R^{\times}$ is a cyclic group 
		of order 4, with generator $\pm i$.
		
		\item Let $p\in \Z$ be a (positive) prime number.  If $p\equiv 3\bmod 4$, show that
		$p$ is prime in $\Z[i]$ and that $\Z[i]/(p)$ is a finite field of characteristic $p$ which, as a vector space over
		$\F_p$, has dimension 2.  
		
		If $p=2$ or $p\equiv 1\bmod 4$, prove that $p$ is not prime 
		in $\Z[i]$, but is the norm of a prime $\mathfrak{p}\in \Z[i]$ with
		$\Z[i]/(\mathfrak{p})$ isomorphic to the finite field $\F_p$.
		Conclude that $p\in \Z$ can be written as
		the sum of two integer squares if and only if $p=2$ or $p\equiv 1\bmod 4$.  
		
	\end{enumerate}
		
\end{enumerate}
\end{document}
