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\title {Sample Algebraic Number Theory Questions}
\author {Thomas R. Shemanske}
\date{June 2007}
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Number Theory Questions}
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% \dedication{This is dedicated to the one I love}
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\section{Introduction}
Graduate students often ask what they should study in preparation for
a qualifying exam. The answer is of course that definitions,
statements of theorems, examples and sketches of proofs of major
theorems are the norm in a qualifying exam. However, this information
is often not perceived by the student in the same way as it is
intended by the faculty member.
To address this disparity, I have included below a number of sample
questions to help you prepare for the algebraic aspect of the number
theory qualifying exam. These questions reflect only some of my own
prejudices, and are not meant to reflect questions which other
examiners might ask. These questions are in no sense intended to
reflect a comprehensive review of the material, but should give you a
good idea of the type and depth of question which you might be
expected to answer. In particular, if you are not comfortable with
the vast majority of the material reflected by these questions, you
are not ready to take the qual.
In the exam itself, expect to be asked questions like those below, but
also some which go beyond the bounds of your knowledge, and possibly
outside the bounds of the syllabus. It is by probing the boundaries of
your knowledge that we ascertain the depth of your knowledge. You are
not expected to know the answer to everything we ask, but you are
expected to know a majority.
Caveat: These problems have not really been proofread yet, but I
wanted them available ASAP.
\section{Global Theory}
\begin{enumerate}
\item Define the ring of integers of a number field. Give a
characterization of the ring of integers of quadratic and
cyclotomic extensions of $\Q$. Prove the result for quadratic
extensions.
\item What is an integral basis? What is the norm of an ideal and why
is it finite?
\item Define the terms ramification index and inertia degree. Which
primes ramify in quadratic and cyclotomic extensions of $\Q$?
\item Let $L/K$ be an extension of number fields, $\p \subset \O_K$ a
prime. Factor $\p\O_L$ as a product of primes $\P_1^{e_1} \cdots
\P_r^{e_r}$. What is the relation of the $e_i$,$r$ and $[L:K]$. If
$L/K$ is Galois, how does this affect things and why? Suppose $\O_L
= \O_K[\alpha]$. How would you determine the $\P_i$?
\item Let $p \equiv 1\pmod 4$ be a prime, and let $K = \Q(\sqrt p)$.
Determine all primes of $\Q$ which ramify in $K$. Determine
congruence conditions which describe all primes $q$ which split
(completely) in $K$, and congruence conditions for those primes
which are inert.
\item Let $p \equiv 1\pmod 4$ be a prime, $K = \Q(\sqrt p)$, and $L
= \Q(\zeta_p)$ be the $p$th cyclotomic extension. Show that $K
\subset L$. Let $q$ be an odd prime which satisfies $q^r \equiv -1
\pmod p$ for some odd positive integer $r$. Determine the factorization
(type) of $q\O_K$ and $q\O_L$. How do the primes which occur in the
factorization of $q\O_K$ factor in $\O_L$? Work this out explicitly
with $p=13$ and $q = 17$.
\item Let $L/K$ be a Galois extension of number fields, $\p \subset
\O_K$ a prime. Let $\P\subset \O_L$ be a prime lying above $\p$.
Describe the decomposition and inertia groups associated to $\p$ and
$\P$. Determine the order of the decomposition group. If $\P'$ is
another prime of $L$ lying above $\p$, how are the decomposition
groups $D(\P\mid \p)$ and $D(\P'\mid \p)$ related? Suppose that
$\p$ is unramified in $L$. Describe how to construct the Frobenius
map $\ds\left[\frac{L/K}{\P}\right]$. Show that for abelian
extensions, the Frobenius map depends only on $\p$ and not on $\P$.
\item Let $K = \Q(\zeta_{13})$, $\p = 17\Z$, $\P$ any prime of $K$
lying above $\p$ and $D$ and $T$ the associated decomposition and
inertia groups. Let $K_D$ and $K_T$ be the associated fixed fields.
Determine them.
\item Show that $\Q(\sqrt{-5})$ has class number 2 using the
Minkowski bound.
\end{enumerate}
\section{Local Theory}
\begin{enumerate}
\item Define a valuation on a field. Characterize archimedean and
non-archimedean valuations. What are equivalent valuations?
\item What is Ostrowski's theorem?
\item Describe the $p$-adic numbers and integers. Characterize the
$p$-adic numbers as Laurent series in $p$. Describe the $p$-adic
integers in terms of Laurent series in $p$ and in terms of the
$p$-adic valuation on $\Q_p$. Show that $\Z_p$ is a discrete
valuation ring. Characterize all the ideals of $\Z_p$.
\item Determine all the archimedean valuations on $\Q(\root 3 \of 2)$.
\item For an extension of number fields $L/K$ and $\p$ a prime in
$\O_K$, describe the normalized valuation $|\cdot |_\p$ on $K$.
Describe all finite extensions of $K_\p$, their valuations, and
degrees.
\item Let $p$ be an odd prime in $\Z$. Use Hensel's lemma to prove
there are precisely three quadratic extensions of $\Q_p$.
\end{enumerate}
\end{document}