Algebraic Number Theory
Prof. Paul Gunnells, LGRT 1115L, 413.545.6009, gunnells at math dot umass
An algebraic number field is a field obtained by
adjoining to the rational numbers the roots of an irreducible rational
polynomial. Algebraic number theory is the study of properies of such
fields. This course will cover the basics of algebraic number theory,
with topics to be studied possibly including the following: number
fields, rings of integers, factorization in Dedekind domains, class
numbers and class groups, units in rings of integers, valuations and
local fields, and zeta- and L-functions.
Algebraic Number Theory by James Milne. Freely available online. This is a very
polished textbook that covers all the main topics in algebraic number
theory. The only serious omission is zeta and L-functions, but they
are discussed in his notes on class field theory. (One might argue,
as Chevalley did, that such objects should remain unmentioned in
algebraic number theory and class field theory, but we will adopt a
more inclusive stance.)
- Cassels, Frohlich, Algebraic Number Theory (Proc. Instructional
Conf., Brighton, 1965). A standard reference, with expositions of
many topics, including local/global fields, cohomology of groups,
class field theory, towers of class fields, Hecke L-functions and
their functional equations, and a fun historical section. Very
challenging for the novice, but it's all there. Recently back in
print. Unsophisticated web searching finds
- Borevich, A. I.; Shafarevich, I. R. Number theory. Translated from
the Russian by Newcomb Greenleaf. Pure and Applied Mathematics,
Vol. 20 Academic Press, New York-London 1966. A classic. Very
hands on. Lots of examples.
- Manin, Yuri Ivanovic; Panchishkin, Alexei A. Introduction to modern
number theory. Fundamental problems, ideas and theories. Translated
from the Russian. Second edition. Encyclopaedia of Mathematical
Sciences, 49. Springer-Verlag, Berlin, 2005. Many precise
definitions, but few complete proofs. Gives a wide overview of the
- Koch, H. Algebraic number theory. Translated from the 1988 Russian
edition. Reprint of the 1992 translation. Springer-Verlag, Berlin,
1997. The companion volume to Manin-Panchishkin. Similar in
style. An excellent way to see the whole subject in the large
without getting bogged down in the details.
- Frohlich, A.; Taylor, M. J. Algebraic number theory. Cambridge
Studies in Advanced Mathematics, 27. Cambridge University Press,
Cambridge, 1993. Emphasizes the role of valuations.
- Neukirch, Jürgen Algebraic number theory. Translated from the 1992
German original and with a note by Norbert Schappacher. With a
foreword by G. Harder. Grundlehren der Mathematischen Wissenschaften
[Fundamental Principles of Mathematical Sciences],
322. Springer-Verlag, Berlin, 1999. Excellent book. Lots of
material, including class field theory. Nice treatment of number
fields from the point of view of Arakelov geometry. Zeta and
L-functions too (the approach complements the material in
Cassels-Frohlich). Full details of just about everything.
- Lang, Serge Algebraic number theory. Second edition. Graduate Texts
in Mathematics, 110. Springer-Verlag, New York, 1994. A classic.
- Marcus, Number Fields. Springer-Verlag, New York, 1977. Another
classic. Very hands on, with many detailed examples and exercises.
Highly recommended by certain number theorists for self-study.
- Rodriguez Villegas, Fernando. Experimental number theory. Oxford
Graduate Texts in Mathematics, 13. Oxford University Press, Oxford,
2007. Shows how to use computation to explore number theory and to
formulate conjectures. Similar in spirit to many examples done in
class (indeed, I learned tons of number theory from Fernando by
having personal demonstrations of these ideas at various conferences
and cafes). Very inspiring. Highly recommended.
Algebraic number theory is a central topic; there are many freely
available sets of lecture notes floating around, in addition to
Milne's. Here are a few I know about. I haven't read them, so can't give
- Pari-GP is one of the standard
tools for computations in number theory. Free, easy
to install, easy to use. I use it all the time for quick
computations, even those having nothing to do with number theory.
Under active development with a large user group.
- SAGE is a free computational algebra
system that includes Pari-GP as a subset (as well as many other free
software packages). Under active development with a large user
- Magma is software package
designed for computations in algebra, number theory, algebraic
geometry and algebraic combinatorics. The philosophy of this system
is rather different from that of Pari-GP. Like SAGE, it's much more
"object-oriented." I don't use it much myself but know many that do
and swear by it. Under active development with a large user group.
- KANT/KASH is another
program that has a lot of the functionality of Pari, and offers more
functionality in some other areas (like in dealing with finite
fields and algebraic function fields). Free, easy to install, easy
to use. Not sure if it's still in active development.
Here is the list of problems (it will be updated every
so often). Some problems might benefit from computer assistance; I
recommend Pari-GP, SAGE, or Magma (the latter is not free but should
be freely available to you as a grad student in our department).
Here are some GP scripts that have been used in prior incarnations of
You might also want to have a look at the Bordeaux database of
of low degree for some examples to play with.
- uniq.gp. Removes duplicate elements from a sorted vector.
- factsig.gp. Factors a polynomial mod p and returns the degrees of the irreducible factors.
- autorder.gp. Computes the order of a Galois automorphism.
- frob.gp. Computes the Frobenius conjugacy class in a Galois extension.
The grades for this course will be based equally on class
participation and homework problems.
Revised: Tue Aug 29 11:47:30 EDT 2017
gunnells at math dot umass dot edu