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\noindent{\Large{\textsc{Modern Algebra II\hfill Assignment 6.2  \\ Fall 2019 \hfill Due October 23}}}

\large

\vspace{1cm}

\begin{exercise}
Let $R$ be a commutative ring and let $A, B, C \subset R$ be ideals.

\begin{itemize}

\item[\bf a.]  If $A+B = R$, prove that $A^m + B^n = R$ for all $m, n \in \N$.  [{\em Suggestion:} This follows easily from Exercise 3.3.3, but can also be proven directly.]

\item[\bf b.]  If $A$ and $B$ are coprime and $AB = C^n$ for some $n \in \N$, prove that there are coprime ideals $A_0$ and $B_0$ so that $A=A_0^n$, $B = B_0^n$ and $A_0 B_0 = C$.  [{\em Suggestion:} Take $A_0 = A+C$ and $B_0 = B+C$.]

\end{itemize}
\end{exercise}

\begin{exercise}
Let $R$ be a PID, $a, b, c \in R$ and $n \in \N$.   If $a$ and $b$ are coprime\footnote{We say $a, b \in R$ are coprime provided $(a)$ and $(b)$ are coprime ideals.} and $ab = c^n$, prove that there exist coprime $a_0, b_0 \in R$ and $u, v, w \in R^{\times}$ so that $a = ua_0^n$, $b = vb_0^n$ and $a_0 b_0 = w c$.
\end{exercise}

\begin{exercise}
Let $a, b \in \Z$.  Prove that if $b$ is even and $\gcd(a,b) = 1$, then $(a+bi,a-bi) = \Z[i]$.
\end{exercise}

\begin{exercise}
A {\em Pythagorean triple} is a 3-tuple $(a,b,c) \in \Z^3$ so that $a^2 + b^2 = c^2$.  A Pythagorean triple is called {\em primitive} if $\gcd(a,b,c)=1$.  

\begin{itemize}

\item[\bf a.]  If $(a,b,c)$ is a primitive Pythagorean triple, prove that $a$ and $b$ have opposite parity.  [{\em Suggestion:} Work modulo 4 and argue by contradiction.]

\item[\bf b.]  Show that if $(a,b,c)$ is a primitive Pythagorean triple and $b$ is even, then there exist $m, n \in \Z$ so that $\gcd(m,n)=1$ and 
\[
\begin{aligned}
a & = m^2 - n^2,\\
b & = 2mn,\\
c & = \pm(m^2+n^2).
\end{aligned}
\]

\item[\bf c.]  Prove that $m$ and $n$ in part {\bf b} must have opposite parity.  Conclude that the even member of a primitive pythagorean triple is always divisible by 4.

\end{itemize}

\end{exercise}



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