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Prerequisites

The preprequisites for this course are math 122 and 123, i.e.  

Course Description

Number fields are finite extensions of the field of rational  
numbers. They have been extensively studied for about 200 years.  
Special examples can be found in the work of Gauss from the early  
part of the 19th century, while the first systematic treatment was  
given by Kummer in the middle of that century. Gauss was motivated  
by his desire to extend his celebrated law of quadratic  
reciprocity, but much early impetus also came from the study of  
Diophantine equations i.e. the search for rational solutions of  
polynomial equations.

Today number fields continue to play a central role in number  
theory. They are both fascinating in their own right and essential  
tools for other work. They are rather concrete objects, with which  
one can make explicit calculations. At the same time they have a  
beautiful abstract structure. Gauss' quest for higher reciprocity  
laws was to a large degree realised with the development of class  
field theory in the first half of the twentieth century. More  
recently much effort has been put into developping even more  
general, `non-abelian' reciprocity laws. This is often referred to  
as the `Langlands program'. Moreover the modern study of  
Diophantine equations would be unrecognizable without number fields.

In this course we introduce the basic concepts related to number  
fields and learn to manipulate them. We will not venture into more  
advanced topics like class field theory or Galois cohomology,  
although those completing math 129 will be in a position to begin  
the study of these subjects. We will take an approach through the  
study of valuations rather than the more classical approach through  
the study of ideals. I feel this is better suited to more advanced  
work in the area.

Topics covered will include: valuations, completions, local fields,  
ramification, rings of integers, ideals, unique factorization of  
ideals, finiteness of class group, structure of unit group,  
Frobenius elements, weak approximation, adeles, and ideles.

Syllabus

 The following syllabus is approximate. Timings may change.

Weeks 1 and 2: Definitions of valuations and absolute values,  
examples and basic properties. Rings of integers and residue  
fields. Weak approximation. Classification of absolute values on  
the rationals. Completion.

Weeks 3-6: Study of complete fields. Hensel's lemma and  
applications. Norms on finite dimensional vector spaces over  
complete fields. Extending valuations to purely transcendental  
extensions. Newton polygons and factorizing polynomials over  
complete fields. Extending valuations to algebraic extensions of  
complete fields. Krasner's lemma. Unramified extensions, tamely  
ramified extensions, wildly ramified extensions. The (lower)  
ramification filtration on the Galois group. The different. The  
local Kronecker-Weber theorem.

Second half of the semester: My plans are less definite. I will  
certainly cover extending valuations in the incomplete algebraic  
case and then turn to number fields. Here I will discuss the  
product formula, ideals and divisors, the class group, the unit  
group, the Kronecker-Weber theorem and some applications to  
Diophantine equations, including the Hasse principle for ternary  
quadrics over the rationals.

Books

There are many good books on number fields. In this course we will  
most closely follow:

J.W.S. Cassels, ``Local Fields'', Cambridge Univ Press 1986, ISBN 0  
521 31525 5.

Other books that I found useful when I was learning the subject  
include:

Stewart and Tall , `` Algebraic Number Theory'', (well written but  
somewhat elementary for this course).

Marcus, ``Number Fields'', (an excellent source of exercises, but  
it takes a different approach from us).

Lang, ``Algebraic number theory'', (fast paced and covers far more  
material than we will in this course).

Cassels and Frohlich (eds), ``Algebraic number theory'', (excellent  
for a second course in algebraic number theory).

However I would recommend browsing in the library, as there are  
many other choices, and I am not personally familiar with them all.




Homework and Exams

There will be approximatelyweekly homeworks. Students are  
encouraged to discuss the homework together, but must write up  
their solutions by themselves. There will also be one midterm  
(probably take home) and a final (a 3 hour traditional exam). The  
grade will be determined as 40% homework, 20% midterm and 40% final.