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\homework{18: More Complex Numbers}{Due Friday at 11:30am, Finals Week}

Solve \textbf{three} of the following \textbf{six} problems.  Also, this set is \textbf{extra-credit!}  This set can be submitted by email if you can't turn it in to my office.  Have fun!


\begin{enumerate}
\item Factor 1500 into primes over the Gaussian integers $\mathbb{Z}[i]$.
\item Prove that there are infinitely many Gaussian integer primes.
\item Let $\omega$ denote the complex number $e^{2\pi i / 3}$, which on the past HW you proved was a third root of unity (that is, $\omega^3 = 1$).  Consider the set $\mathbb{Z}[\omega]$ of the \textbf{Eisenstein integers}, defined as follows:
\begin{align*}
\mathbb{Z}[\omega] = \{a + b \omega ~|~ a, b \in \mathbb{R}\}.
\end{align*}

\begin{enumerate}
\item Prove that the Eisenstein integers are closed under multiplication and addition.
\item Let $N(a + b\omega)$ denote the square of the distance of $a+b\omega$ from the origin in $\mathbb{C}$.  Prove that $N(a+b\omega) = a^2 + b^2 - ab$.
\item Show that $\pm 1, \pm \omega, \pm (1+\omega)$ are the only three Eisenstein integers $z$ for which $N(z) = 1$.
\end{enumerate}
\item Create a version of the Euclidean algorithm that works for the Eisenstein integers.  Use it to prove that in the Eisenstein integers, all elements can be factored uniquely into Eisenstein-integer primes.
\item Use the Eisenstein integers to find all integer solutions to the equation $y^2 = x^3 + 1$.
\item Prove that unique factorization does not hold for the Hurwitz integers: that is, find some $\tau = a+bi+cj+dk \in \mathcal{H}$ with two distinct factorizations into different sets of irreducible elements. 
\end{enumerate}


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