Math 7470 Fall 2014
Assignments
Learning Outcome
Class:
- Math 7470
- 328 State Hall
- TTh 11:45 - 1:10
- Text:
Gerald J. Janusz, Algebraic Number Fields
(American Math Soc., GSM Vol 7, ISBN: 0-8218-0429-4 )
Professor:
Schedule:
First class: Thursday, 28 August.
Thanksgiving holiday: Thursday, 27 November.
Last class: Thursday, 4 December.
You will
be responsible for material covered in class, whether it is in the book
or not.
Regular homework problems will be assigned.
Class members will be asked to present problems and other material in class.
Final Exam: there will be no final exam.
Learning Outcomes: You will be able to compute many interesting
invariants of algebraic number fields.
Software:
Sage has extensive number theoretic capabilities and is free to download.
Other mathematical software, such as MAGMA and GAP, can do some
number theoretic calculations as well.
Bibliography:
The following books were close contenders in my search
for a text. They are all worth looking at.
- Algebraic Number Theory,
A. Frohlich and M. J. Taylor.
An excellent text that I chose not
to adopt mainly because I felt it would take too much time to cover the
introductory material at the level they present. Their chapter on fields of low
degree nearly made me choose it anyway.
- Number Fields,
Daniel Marcus.
An idiosyncratic approach that has a lot to recommend it.
Many results are given simpler, clearer, proofs than is common.
- A Brief Guide to Algebraic Number Theory,
H. P. F. Swinnerton-Dyer.
Beautiful, but perhaps too terse and brief
for our purposes.
- A Classical Introduction to Modern Number Theory,
Kenneth Ireland and Michael Rosen.
Perhaps the most standard choice for this
course, but it simply didn't inspire me.
Here are two books which will be useful.
They will be on the bookshelf of anyone
in the field.
- Algebraic Number Theory,
Ed. by J. W. S. Cassells and A. Frohlich.
This is the standard reference
from which my generation learned this material. The level jumps around quite a lot
since it was written by many authors. They were among the leading researchers in the
field at the time the book was written.
- Algebraic Number Theory,
Jurgen Neukirch.
Beautiful and elegant, but it would be beyond my reach
to teach an
introductory course from it. Still, I highly recommend that anyone read the
beginning sections to see how cleanly and quickly interesting material can be reached.
In addition, these expository books are marvelous ways to get
a broad view of the subject. They contain an enormous number of
interesting examples.
- Number Theory I: Fermat's Dream, by Kato, Kurokawa and Saito.
AMS Trans. Math. Mono. V. 186.
- Number Theory II: Introduction to Class Field Theory,
by Kato, Kurokawa and Saito.
AMS Trans. Math. Mono. V. 240.
This is a very short and idiosyncratic bibliography. The field is
enormous and there are now many other books that are worth consulting.
(Text = Janusz)
You will be able to compute many interesting invariants of algebraic
number fields.
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