A Course In Algebraic Number Theory

Content URL: Link To Content
This is a text for a basic course in algebraic number theory, written in accordance with the following objectives. 

1. Provide reasonable coverage for a one-semester course. 
2. Assume as prerequisite a standard graduate course in algebra, but cover integral extensions and localization before beginning algebraic number theory. For general algebraic background, see my online text “Abstract Algebra: The Basic Graduate Year”, which can be downloaded from my web site www.math.uiuc.edu/∼ r-ash/ The abstract algebra material is referred to in this text as TBGY. 
3. Cover the general theory of factorization of ideals in Dedekind domains, as well as the number field case. 
4. Do some detailed calculations illustrating the use of Kummer’s theorem on lifting of prime ideals in extension fields. 
5. Give enough details so that the reader can navigate through the intricate proofs of the Dirichlet unit theorem and the Minkowski bounds on element and ideal norms. 
6. Cover the factorization of prime ideals in Galois extensions. 
7. Cover local as well as global fields, including the Artin-Whaples approximation theorem and Hensel’s lemma.  

A Computational Introduction to Number Theory and Algebra

Content URL: Link To Content
Number theory and algebra play an increasingly significant role in computing and communications, as evidenced by the striking applications of these subjects to such fields as cryptography and coding theory. My goal in writing this book was to provide an introduction to number theory and algebra,with an emphasis on algorithms and applications, that would be accessible to a broad audience. In particular, I wanted to write a book that would be accessible to typical students in computer science or mathematics who have a some amount of general mathematical experience, but without presuming too much specific mathematical knowledge. 

Prerequisites. 
The mathematical prerequisites are minimal: no particular mathematical concepts beyond what is taught in a typical undergraduate calculus sequence are assumed. 

The computer science prerequisites are also quite minimal: it is assumed that the reader is proficient in programming, and has had some exposure to the analysis of algorithms, essentially at the level of an undergraduate course on algorithms and data structures. 

Even though it is mathematically quite self contained, the text does presuppose that the reader is comfortable with mathematical formalism and has some experience in reading and writing mathematical roofs. Readers may have gained such experience in computer science courses such as algorithms, automata or complexity theory, or some type of “discrete mathematics for computer science students” course. They also may have gained such experience in undergraduate mathematics courses, such as abstract or linear algebra—these courses overlap with some of the material presented here, but even if the reader already as had some exposure to this material, it nevertheless may be convenient to have all of the relevant material easily accessible in one place, and moreover, the emphasis and perspective here will no doubt be different than in a typical mathematics course on these subjects.  

An Introduction to the Theory of Numbers

Content URL: Link To Content
These lectures are intended as an introduction to the elementary theory of numbers. I use the word "elementary" both in the technical sense|complex variable theory is to be avoided - and in the usual sense - that of being easy to understand, I hope. 

I shall not concern myself with questions of foundations and shall presuppose familiarity only with the most elementary concepts of arithmetic, i.e., elementary divisibility properties, g.c.d. (greatest common divisor), l.c.m. (least common multiple), essentially unique factorizaton into primes and the fundamental theorem of arithmetic: if p j ab then p | a or p | b. 

I shall consider a number of rather distinct topics each of which could easily be the subject of 15 lectures. Hence, I shall not be able to penetrate deeply in any direction. On the other hand, it is well known that in number theory, more than in any other branch of mathematics, it is easy to reach the frontiers of knowledge. It is easy to propound problems in number theory that are unsolved. I shall mention many of these problems; but the trouble with the natural problems of number theory is that they are either too easy or much too difficult. I shall therefore try to expose some problems that are of interest and unsolved but for which there is at least a reasonable hope for a solution 
by you or me. 

The topics I hope to touch on are outlined in the Table of Contents, as are some of the main reference books. Most of the material I want to cover will consist of old theorems proved in old ways, but I also hope to produce some old theorems proved in new ways and some new theorems proved in old ways. Unfortunately I cannot produce many new theorems proved in really new ways.