8251 Lecture Notes - Fall 2014

Notes will be posted here in the afternoons following the lecture.

Lecture Notes

Date

Topic

#1
Wednesday,
Sept. 3,
Outline of the course, sums of squares
#2
Friday,
Sept. 5,
Algebraic integers
#3
Monday,
Sept. 8,
Trace, norm, discriminant
#4
Wednesday,
Sept. 10,
integral bases
#5
Friday,
Sept. 12,
ideal factorization, Dedekind domains I
#6
Monday,
Sept. 15,
ideal factorization, Dedekind domains II
#7
Wednesday,
Sept. 17,
fractional ideals, class group, lattices
#8
Friday,
Sept. 19,
Minkowski's theory of lattices
#9
Monday,
Sept. 22,
Finiteness of class group, start of discussion on units
#10
Wednesday,
Sept. 24,
End of Dirichlet unit theorem, regulator
#11
Friday,
Sept. 26,
Factoring primes in Dedekind domains
#12
Monday,
Sept. 29,
Factoring primes in Dedekind domains II
#13
Wednesday,
Oct. 1,
Factoring primes in Galois extensions
#14
Friday,
Oct. 3,
Using SAGE to do computational algebraic number theory
#15
Monday,
Oct. 6,
Cyclotomic fields I
#16
Wednesday,
Oct. 8,
Cyclotomic fields II
#17
Friday,
Oct. 10,
Localization
#18
Monday,
Oct. 13,
Units and class groups in localizations
#19
Wednesday,
Oct. 15,
one dimensional Noetherian domains
#20
Friday,
Oct. 17,
Geometric intuition
#21
Monday,
Oct. 20,
Affine schemes for 1-dimensional domains
#22
Wednesday,
Oct. 22,
Function fields
#23
Friday,
Oct. 24,
Introduction to valuation theory
#24
Monday,
Oct. 27,
completeness of p-adic numbers
#25
Wednesday,
Oct. 29,
classification of valuations
#26
Friday,
Oct. 31,
completions of number fields
#27
Monday,
Nov. 3,
non-archimedean valuations, Hensel's lemma
#28
Wednesday,
Nov. 5,
Extensions of absolute values
#29
Friday,
Nov. 7,
Intro to local fields
#30
Monday,
Nov. 10,
Unit theorems
#31
Wednesday,
Nov. 12,
Logarithm in p-adic fields
#32
Friday,
Nov. 14,
Extensions of Henselian fields
#33
Monday,
Nov. 17,
The structure of unramified extensions
#34
Wednesday,
Nov. 19,
Extension of valuations in number fields
#35
Friday,
Nov. 21,
The fundamental identity of local extensions
#36
Monday,
Nov. 24,
Galois theory of local extensions
#37
Monday,
Dec. 1,
Formalism of profinite groups
#38
Wednesday,
Dec. 3,
Formalism of valuations
#39
Friday,
Dec. 5,
The reciprocity map
#40
Monday,
Dec. 8,
local class field theory and Kronecker-Weber theorems
#41
Wednesday,
Dec. 10,
global class field theory

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