Homework number 5 - Due Friday, November 2, 2007

hw5.pdf

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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
  you intend to do your final project about.

\end{enumerate}


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