Pries, Spring 2016, Algebraic Number Theory Syllabus
Week
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Starts Topics
Introduction to number fields
1/20
Quadratic fields, quadratic reciprocity
1/25
Cyclotomic fields, number fields, norm and trace maps, guestMW
2/1
Integral closure, discriminant
Class groups
2/8
Unique factorization, ideal class group
2/15
Norms of ideals, Minkowski theory
2/22
Class number and Dirichlet’s Unit Theorem
Abelian extensions
2/29
Kummer theory, Ray class groups
3/7
Kronecker-Weber theorem, Complex multiplication
spring break
Ramification
3/21
Decomposition and Inertia groups
3/28
Localization, Kummer’s theorem, guestF
4/4
Frobenius automorphism, Artin map
Applications
4/11
Kummer’s proof of Fermat’s Last Theorem for regular primes
4/18
Lattice-based cryptography
4/25
Presentations
5/2
Presentations, guestW
Textbooks: These books contain similar material from different points of view.
Free/open source: (see links from webpage)
* Milne Algebraic Number Theory C 1-6, 8
* Stein Brief Introduction to Classical and Adelic Algebraic Number Theory C 5-6, 10-14
(algorithms MAGMA computing)
* Ash A Course in Algebraic Number Theory
* Ogglier Introduction to Algebraic Number Theory C 1-4
Concrete:
* Ribenboim Classical Theory of Algebraic Numbers C 2, 4-15, 19, 25 (very detailed)
Ireland-Rosen A Classical Introduction to Modern Number Theory C 5, 12, 13 (better for
arithmetic geometry)
Marcus Number Fields
Esmonde/Murty Problems in Algebraic Number Theory C 4-9 (problem based style)
Classical: * Janusz Algebraic Number Fields C 1,3.
Lang Algebraic Number Theory C 1-6.
Cassels/Frolich Algebraic Number Theory
Cohn Advanced Number Theory
Harder:
* Cohen Number Theory I C 2-3.
Neukirch Algebraic Number Theory C 1.