Algebraic Number Theory


Prof. Paul Gunnells, LGRT 1115L, 413.545.6009, gunnells at math dot umass dot edu.

Office Hours



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.)

Other books

Online references

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 comments.



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 this class. You might also want to have a look at the Bordeaux database of number fields of low degree for some examples to play with.


The grades for this course will be based equally on class participation and homework problems.

Revised: Tue Aug 29 11:47:30 EDT 2017
Paul Gunnells
gunnells at math dot umass dot edu