Algebraic Number Theory
Math 523 - Spring 2018
Announcements
The semester is over. Have a nice summer.
Exercise Sets
Exercise Set 11: Marcus Chapter 5, Exercises 6-25.
Exercise Set 12: Marcus Chapter 5, Exercises 34-41, 46-48.
Grading: Exams & Homeworks
Schedule of Lectures and Office Hours
The schedule of lectures:
Tuesday 8:40-10:30 and Wednesday 9:40-10:30 in M231.
I have office hours after the lectures:
Tuesday and Wednesday between 10:40-12:00.
If these office hours are not suitable for you, then you can stop by any time. If I am suitable, I will be happy to talk about number theory.
Textbooks and Tentative Course Outline
There are four textbooks. However we will mostly use the first two.
1) Marcus, Number Fields,
2) Stewart & Tall, Algebraic Number Theory and F.L.T., 3rd edition,
3) Washington, Introduction to Cyclotomic Fields, 2nd edition and
4) Cox, Primes of the Form x2 + ny2.
Find a tentative outline below for the whole semester. For each week, we will attempt to cover the indicated pages.
(Feb 13 - Feb 14) Introduction. Pythagorean triples and Gaussian integers. [1, 1-11].
(Feb 20 - Feb 21) Algebraic background. [2, 9-34].
(Feb 27 - Mar 28) Algebraic numbers. Algebraic integers. Integral bases. [2, 35-48].
(Mar 06 - Mar 07) Traces. Norms. Rings of integers. [2, 49-59].
(Mar 13 - Mar 14) Cyclotomic fields. [1, 17-19, 27, 30-36].
(Mar 20 - Mar 21) Factorization into irreducibles. Euclidean domains. [2, 79-93].
(Mar 27 - Mar 28) Two Diophantine equations. Dedekind domains. [2, 94-98] [1, 55-62]
(Apr 03 - Apr 04) Prime factorization of ideals. [1, 62-82].
(Apr 10 - Apr 11) Galois theory. [1, 258-264 Appendix]
Midterm - April 10
(Apr 17 - Apr 18) Galois theory applied to prime decomposition. [1, 98-114]
(Apr 24 - Apr 25) Class group and class number. Minkowski's constant. [1, 130-140]
(May 01 - May 02) Class group examples.
(May 08 - May 09) Dirichlet's unit theorem. Pell's equation. [1, 141-146]
(May 15 - May 16) The first case of Fermat's Last Theorem. [3, 1-8]
(May 22) Class number formula. Primes of the form x2+ny2. [3] and [4]
Final - May 23
PARI / GP
The software PARI / GP is very simple to learn and
extremely strong to do computations with.