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# Homework number 5 - Due Friday, November 2, 2007

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\Large\bf Homework 5 for Math 581F\\
Due FRIDAY November 2, 2007
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Each problem has equal weight, and parts of problems are worth the
same amount as each other.  This homework assignment has three
problems that are quite easy, one that is very open ended,
and two that might be difficult for you.

\begin{enumerate}

\item Let $\mathfrak{p}$ be one of the  prime ideals that divides
$2$ in the ring of integers of $\QQ(\sqrt{-13})$.  Describe
the abstract abelian group structure of the additive group
of $\mathfrak{p}^{2006} / \mathfrak{p}^{2007}$.

\item (*) Give an example of an order $\O$ in the ring of integers of
a number field and an ideal $I$ such that~$I$ cannot be generated by
$2$ elements as an ideal.  Does the Chinese Remainder Theorem hold
in $\O$?  [The (*) means that this problem is more difficult than
usual.]

\item Let $a_1 = 1+i$, $a_2 = 3+2i$, and $a_3 = 3+4i$ as elements of
$\Z[i]$.
\begin{enumerate}
\item Prove that the ideals $I_1=(a_1)$, $I_2=(a_2)$, and $I_3=(a_3)$
are coprime in pairs.
\item Compute $\#(\Z[i]/(I_1 I_2 I_3))$.

\item Find a single element in $\Z[i]$ that is congruent to~$n$ modulo
$I_n$, for each $n\leq 3$.  [Note: Sage doesn't have CRT over rings
of integers yet, so you'll have to either do this problem partly by
hand, which is definitely possible, or use another program such as PARI or
Magma.]
\end{enumerate}

\item
Let $\p$ be a prime ideal of $\O_K$, and suppose that $\O_K/\p$
is a finite field of characteristic $p\in\Z$.  Prove that there is
an element $\alpha\in\O_K$ such that $\p=(p,\alpha)$.  This
justifies why we can represent prime ideals of $\O_K$ as pairs
$(p,\alpha)$, as is done in \sage.

\item (*) Use cyclotomic fields to prove that for every $n$ there
exists a number field $F$ of degree $n$ in which $2$ splits
comletely, i.e., factors as a product of $n$ distinct primes in the
ring of integers of $F$.  Conclude that for every $d$ there exists a
number field whose ring of integers requires at least $d$
generators.  [Hint: See
{\tt http://wstein.org/129-05/challenges.html}, where a solution is
given that is slightly sketchy, and that depends on learning some
things about cyclotomic fields that I haven't covered in class.  You
may use theorems from other books or articles that we haven't proved
in class or that aren't proved in the course textbook.  Just give a
precise citation for the theorem.]

\item Give a clear description that is less than a page long of what