\documentclass[12pt]{article} \usepackage{amsfonts} \usepackage{amsmath, amsthm, amssymb} \usepackage{myfullpage} %%%% If you want 1 inch margins \pagestyle{myheadings} %------------------------------------------- % Begin Topmatter infomation %------------------------------------------- \title {Sample Algebraic Number Theory Questions} \author {Thomas R. Shemanske} \date{June 2007} %\predate %With time and preliminary version appended \markboth{Sample Algebraic Number Theory Questions} {Sample Algebraic Number Theory Questions} %------------------------------------------- % End Topmatter infomation %------------------------------------------- %------------------------------------------- % Begin Local Macros %------------------------------------------- \newcommand{\Z}{{\mathbb Z}} \newcommand{\Q}{{\mathbb Q}} \newcommand{\C}{{\mathbb C}} \renewcommand{\O}{{\mathcal O}} \renewcommand{\P}{{\mathfrak P}} \newcommand{\p}{{\mathfrak p}} \newcommand{\ds}{\displaystyle} %------------------------------------------- % End Local Macros %------------------------------------------- \emergencystretch=20pt \begin{document} \maketitle %------------------------------------------- % Other Topmatter Information %------------------------------------------- % \dedication{This is dedicated to the one I love} % \subjclass {Primary 11Fxx; Secondary 11Fxx} % \keywords{Maximal Order, Central Simple Algebra, Bruhat--Tits % Building} % \begin{abstract} % This is a great paper. Read no further, because I don't want you to % hurt yourself, but if you can't help yourself, better strap in. It % gets bumpy from here on in. % \end{abstract} %------------------------------------------- % End Other Topmatter Information %------------------------------------------- % \def\today{\ifcase\month\or January\or February\or March\or April\or % May\or June\or July\or August\or September\or October\or November\or % December\fi\space\number\day, \number\year} \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}