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\begin{document}

\begin{center}
\bf
Math 254A, UC Berkeley, Fall 2001 (Kedlaya) \\
Final Examination \\
Due in my mailbox Monday, December 10 by 5 PM
\end{center}

\head{Instructions}
Please submit solutions to at least eight of the following problems; 
your best eight problems will be counted. You \emph{may}:
\begin{itemize}
\item Consult Neukirch, your own lecture notes, any handouts distributed in
class, your graded problem sets, or the Magma documentation.
\item Invoke the statement of any result from Neukirch, from lecture,
or from the problem sets (submitted or not)
without proof, as long as you specify where it
came from.
\item Consult me or Frank with questions.
\item Use Magma (or a computer algebra system of your choice)
where specified.
\end{itemize}
You \emph{may not}:
\begin{itemize}
\item
Discuss the exam questions with anyone other than me or Frank,
until after you have submitted the exam.
\item
Use Magma or other computational device where not directed to do so.
\item 
Consult any reference, print or electronic, other than those mentioned above.
\end{itemize}

\head{Problems}

Throughout the problems, let $\zeta_n$ denote a primitive $n$-th root of
unity.
\begin{enumerate}
\item
  \begin{enumerate}
\item[(a)] Prove that the field $\QQ(\zeta_{13})$ contains a unique
subfield $K$ of degree 4 over $\QQ$, generated by
$x = \zeta_{13} + \zeta_{13}^3 + \zeta_{13}^9$.
\item[(b)] Determine the minimal polynomial and the different
of $x$. You may use Magma for this
and the next part.
\item[(c)] Exhibit an element of $\gotho_K$ not contained in $\ZZ[x]$.
(Hint: compute the discriminant of the minimal polynomial of $x$ first
to find a square factor that you can remove.)
  \end{enumerate}
\item 
Let $\alpha$ be an algebraic integer all of whose conjugates in $\CC$ are
real and lie in the interval $[-2, 2]$. Prove that $\alpha = 2 
\cos(2\pi r)$ for some $r \in \QQ$. (Hint: reduce to Kronecker's theorem from
Problem Set 2.)
\item
  \begin{enumerate}
  \item[(a)] Show that the class number of $\ZZ[\frac{1}{2}(1 + \sqrt{-11})]$
 is 1. (Hint: use the bounds from Problem Set 5 to reduce this to a finite
computation.)
\item[(b)] Find all integers $x,y \in \ZZ$ such that $y^2 = x^3 - 11$.
  \end{enumerate}
\item
Find all solutions of the equation
\[
x^3 + 5 y^3 + 25 z^3 - 15xyz = \pm 1
\]
with $x,y,z \in \ZZ$. You may use Magma to search for some small solutions.
(Hint: relate this to the unit group
of $\QQ(\sqrt[3]{5})$, whose ring of integers you may assume is
$\ZZ[\sqrt[3]{5}]$ without proof.)
\item
Let $K$ be a finite unramified extension of $\QQ_p$ with valuation ring
$\gotho_K$, and let $\gothm_K$ be the maximal ideal of $\gotho_K$.
Let $\sigma$ be the unique automorphism of $K$ such that $x^\sigma \equiv
x^p \pmod{\gothm_K}$.
Prove that for any $x \in \gotho_K$ such that $x \equiv 1 \pmod{\gothm_K}$,
there exists $y \in \gotho_K^*$ such that $y^\sigma/y^p = x$. (Hint:
imitate one proof of Hensel's lemma.)
\item
Determine the number and degrees of
the irreducible factors of the polynomial $x^4 + 9x^2-2$ over
$\QQ_2$. (Hint: make sure to apply Hensel's lemma only when valid.)
\item
Let $L$ be a number field whose absolute discriminant is squarefree and let $K$
be a subfield of $L$. Prove that $K = \QQ$. (Hint: use properties of the
relative discriminant to show that $K/\QQ$ is unramified.)
\item
Let $P(x) = x^6 - x^5 - x^4 + x + 1$. You may assume $P(x)$ is irreducible
over $\ZZ$.
\begin{enumerate}
\item[(a)] Compute the discriminant $D$ of $P(x)$ using Magma, and
check that it is $3^3$ times a squarefree number.
\item[(b)] Let $K$ be the normal closure of $\QQ[x]/(P(x))$. Prove that $K$ is
unramified over all primes of $\QQ(\sqrt{D})$ not above $(3)$.
\item[(c)] Prove that $K$ is also unramified over the prime of $\QQ(\sqrt{D})$
above $(3)$. (Hint: determine the decomposition and inertia groups for
a prime of $K$ above $(3)$.)
\end{enumerate}
\item
Let $n$ be a positive integer which is not a prime power.
Prove that $1 - \zeta_n$
is a unit in $\ZZ[\zeta_{n}]$. (Hint: compute $\prod_{i=1}^{n-1}
(1 - \zeta_{n}^i)$ and show that the terms where $i$ is not coprime
to $n$ account for all of the prime factors. For starters, try
the case $n = pq$.)
\item
\begin{enumerate}
  \item[(a)] Prove that the field $\QQ(\zeta_{31})$ contains a
unique subfield $K$ of degree 5 over $\QQ$.
\item[(b)]
  Prove that 2 is unramified and splits completely in $K$.
\item[(c)]
Use (b) to prove that there does not exist $\alpha \in \gotho_K$ such that
$\gotho_K = \ZZ[\alpha]$. (Hint: derive a contradiction from 
the splitting criterion that holds when $\gotho_K = \ZZ[\alpha]$.)
  \end{enumerate}
\item
Neukirch exercise III.2.1:
\begin{enumerate}
\item[(a)]
Let $L/K$ be finite extensions of $\QQ_p$. Prove that $\Disc(L/K)$ is generated
by $\Disc(1, \alpha, \dots, \alpha^{n-1})$, where $\alpha$ runs over all
elements of $\gotho_L$ such that $K(\alpha) = L$.
\item[(b)]
Prove that this statement does not hold for number fields, by showing that
it fails for $K = \QQ$ and $L = \QQ[x]/(P(x))$, where
$P(x) = x^3-x^2-2x-8$.
You may use without proof the fact that the discriminant of $P$ is $-2012$.
(Hint: show that $1, x, (x^2+x)/2$ is an integral basis of $L$.)
\end{enumerate}
\end{enumerate}

\end{document}