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\begin{document}

\begin{center}
\bf
Math 254A, UC Berkeley, Fall 2001 (Kedlaya) \\
Problem Set 2: Number Fields and Ideals \\
Due in class Monday, September 10
\end{center}

\head{Reference} Neukirch, Sections 1.2 and 1.3.

\head{Jargon watch} An ideal of a ring of integers is sometimes called an
\emph{integral ideal} to emphasize that it is not just a fractional ideal.

\head{Conjugates, Trace and Norm}

Please submit all of the following problems.

\begin{enumerate}
\item
(A theorem of Kronecker)
Let $\alpha$ be an algebraic integer all of whose conjugates in
$\CC$ lie in the closed unit disc. Prove that $\alpha$ is a root of unity.
(Hint: consider the powers of $\alpha$.)
\item
Let $x$ be an element of a number field $K$ such that $\Trace(x^n)$
is a rational integer for all $n \in \NN$. Prove that $x$ is an
algebraic integer.
\end{enumerate}

\head{Ideals and Unique Factorization}

Please submit four of the following problems.

\begin{enumerate}
\item
Find an example to show that a proper subring of the ring of integers
of a number field need not have unique factorization of ideals. (Hint:
this can already be done in a quadratic number field.)

\item
Neukirch exercise I.3.3: Let $d$ be squarefree and $p$ a prime number not
dividing $2d$. Let $\gotho$ be the ring of integers of $\QQ(\sqrt{d})$. Show
that $(p) = p\gotho$ is a prime ideal of $\gotho$ if and only if the
congruence $x^2 \equiv d \pmod{p}$ has no solution in rational integers.

\item
Show that $\ZZ[\sqrt{-5}]$ has exactly two ideal classes, the class of
principal ideals and the class containing $(2, 1+\sqrt{-5})$. (Hint:
suppose $\gotha$ is an ideal which is not principal and $\alpha$ is a 
nonzero element of $\gotha$ of minimum norm. Prove that $2\gotha \subseteq
(\alpha)$.)

\item
Neukirch I.3.6: Prove that every ideal of a Dedekind domain can be 
generated by two elements. (Hint: use Neukirch I.3.5: The quotient of
a Dedekind domain by a nonzero ideal is a principal ideal ring.)

\item
Neukirch I.3.8: Let $\gothm$ be a nonzero integral ideal of the Dedekind
domain $\gotho$. Show that in every ideal class of $\gotho$, there exists
an integral ideal prime to $\gothm$.

\item
Let $K$ be any field and let $L$ be a finite separable extension of
the field $K(x)$ of rational functions over $K$. Prove that the integral
closure of $K[x]$ in $L$ is a Dedekind domain. (Hint: imitate the proof of
Theorem~I.3.1 in Neukirch.)

\end{enumerate}

\end{document}

\head{Quadratic and cyclotomic fields}
\begin{enumerate}
\item
Let $r$ and $s$ be relatively prime integers. Prove that the conjugates
of $\exp(2 \pi i r/s)$ in $\CC$ are precisely
the numbers $\exp(2 \pi i m/s)$ for all integers $m$ relatively prime to $s$.
(Hint: to show these all occur, prove that the ring $\QQ[x]/(x^s-1)$ admits
an automorphism sending $x$ to $x^m$.)
\item
Using the previous exercise, write down a formula for the minimal polynomial
of $\exp(2 \pi i/n)$ in terms of the prime factorization of $n$. (Hint:
if you're having trouble, first suppose $n$ is prime, then squarefree.)
Compute some examples using your formula, and use Magma's builtin
function \verb+CyclotomicPolynomial+ to test your computations.
\end{enumerate}