Homework number 4 - Due Friday, October 26, 2007

hw4.pdf

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\Large\bf Homework 4 for Math 581F\\
Due FRIDAY October 26, 2007
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Each problem has equal weight, and parts of problems are worth the
same amount as each other.

\begin{enumerate}

\item
\begin{enumerate}
\item Find by hand  and with proof 
the ring of integers of each of the following two fields: 
$\Q(\sqrt{5})$, $\Q(i)$.  
\item Find the ring of integers of $\Q(x^5+7x+1)$ using a computer. 
\end{enumerate}


\item Let $\O_K$ be the ring of integers of a number field $K$,
and let $p\in\Z$ be a prime number.  What is the cardinality
of $\O_K/(p)$ in terms of $p$ and $[K:\Q]$, 
where $(p)$ is the ideal of $\O_K$ generated by~$p$?

\item Explicitly factor the ideals generated by each of $2$, $3$, and
  $13$ in the ring of integers of $\Q(\sqrt[3]{2})$.  (Thus you'll
  factor three separate integral ideals as products of prime ideals.)
  You may assume that the ring of integers of $\Q(\sqrt[3]{2})$ is
  $\Z[\sqrt[3]{2}]$, but do {\em not} simply use a computer command to
  do the factorizations.

\item
Let $K=\Q(\zeta_{13})$,where $\zeta_{13}$ is a primitive
$13$th root of unity.  Note that~$K$ has ring of integers $\O_K=\Z[\zeta_{13}]$.
\begin{enumerate}
\item Factor $2$, $3$, $5$, $7$, $11$, and $13$ in the ring 
of integers $\O_K$.  You may use a computer.  
\item For $p\neq 13$, find a conjectural 
relationship between the number of prime ideal factors of $p\O_K$
and the order of the reduction of~$p$  in $(\Z/13\Z)^*$.
\item Compute the minimal polynomial $f(x)\in\Z[x]$ of $\zeta_{13}$.
Reinterpret your conjecture as a conjecture that
relates the degrees of the irreducible factors of $f(x)\pmod{p}$ to 
the order of $p$ modulo~$13$.  Does your conjecture 
remind you of quadratic reciprocity?
\end{enumerate}


\item Let $p$ be a prime.  Let $\O_K$ be the ring of integers of a
  number field~$K$, and suppose $a\in \O_K$ is such that
  $[\O_K:\Z[a]]$ is finite and coprime to~$p$.  Let $f(x)$ be the
  minimal polynomial of~$a$.  We proved in class 
that if the reduction $\overline{f}\in\F_p[x]$ of $f$ factors
as 
$$
 \overline{f} = \prod g_i^{e_i},
$$
where the $g_i$ are distinct irreducible polynomials in $\F_p[x]$, then the primes appearing
in the factorization of $p\O_K$ are the ideals $(p,g_i(a))$.
In class, we did not prove that the exponents of these primes in the factorization
of $p\O_K$ are the $e_i$.  Prove this.

\item 
\begin{enumerate}
\item Give an example of a cubic {\em Galois} extension $K$ of $\QQ$.  Use
Sage to factor each prime $p<100$ (or more) in $\O_K$ and record
the number of prime factors of each $p$.
\item Give an example of a cubic {\em non-Galois} extension $K$ of $\QQ$.  Use
Sage to factor each prime $p<100$ (or more) in $\O_K$ and record
the number of prime factors of each $p$.
\item Come up with a more refined conjecture about the proportion
of primes $p$ for which the number of prime factors is $1$, $2$
or $3$ in each of the above two cases. 
\end{enumerate}


\end{enumerate}


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