Monday 
Wednesday 
Friday 
Sept. 7
In Lecture:
 Sept. 9
In Lecture:
 Algebraic numbers, algebraic integers and transcendental numbers
Associated Reading:
 Lang, Chapter I, sections 1.1 and 1.2
 Hardy and Wright, section 11.7
 Samuel, chapter 2, sections 2.1 to 2.5
Video of class
Video of class (downloadable)
 Sept. 11
In Lecture:

Sept. 14
In Lecture:
 Norms and Traces
 Iintegral closures
 Prime ideals and maximal ideals
Associated Reading:
 Lang, Chapter I, sections 1.2, 1.3, 1.4, 1.5.
 Samuel, chapter 2.
Video of class
Video of class (downloadable)
 Sept. 16
In Lecture:
 Integral closures, continued
 Traces and norms, finiteness of integral closures, continued
 Dual bases
Associated Reading:
 Lang, chapter I, sections 1.2, 1.3, 1.4, 1.5
 Lang, chapter 3, section 3.1.
 Samuel, chapter 2.
Video of class
Video of class (downloadable)
 Sept. 18
In Lecture:

Sept. 21
In Lecture:
Associated Reading:
 Samuel, chapter 2, sections 2.7, 2.8.
 Lang, chapter 3, section 3.3.
Video of class
Video of class (downloadable)
 Sept. 23
In Lecture:
 Rings of cyclotomic integers
 Statement of Kronecker Weber Theorem, Hilbert's 12th problem
 Drinfeld modules
 Beginning of Dedekind rings
Associated Reading:
Video of class
Video of class (downloadable)
 Sept. 25
In Lecture:

Sept. 28
In Lecture:
 Fractional ideals of Dedekind rings form a group
 Integral closures of Dedekind rings
 Ideal class groups, elliptic curves, Jacobians of curves
Associated Reading:
 Samuel, chapter 3.
 Samuel, chapter 5, sections 5.1 and 5.2.
 Lang, chapter 1, section 1.6.
 Hartshorne, chapters 1.6 and 2.6.
Video of class
Video of class (downloadable)
 Sept. 30
In Lecture:
 Applications of Dedekind rings: Fermat, Curves over fields and Dedekind rings
 Decomposition of primes
Associated Reading:
 Washington, chapters 1 and 2.
 Hartshorne, chapters 1.6 and 2.6
 Samuel, chapter 5, sections 5.1 and 5.2.
 Lang, chapter 1, section 1.6.
Video of class
Video of class (downloadable)
 Oct. 2
In Lecture:

Oct. 5
In Lecture:
 End of the proof of main result about decomposition of primes
 Discrete valuation rings
 Decomposition of primes in Galois extensions
 Explicit decompositions of primes
Associated Reading:
 Samuel, chapter 5, sections 5.1 and 5.2.
 Lang, chapter 1, section 1.6, 1.7, 1.8.
Video of class
Video of class (downloadable)
 Oct. 7
In Lecture:
 Examples of discrete valuation rings
 Absolute values and completions
Associated Reading:
Video of class
Video of class (downloadable)
 Oct. 9
In Lecture:
 The time and location of the class will be sent by email the night of Oct. 8
Associated Reading:
 Absolute values and completions, continued
Video of class
Video of class (downloadable)

Oct. 12
In Lecture:
Associated Reading:
 Borevich and Shafarevitch, see index for "Ostrowski's Theorem".
 Lang, chapter 2.
Video of class
Video of class (downloadable)
 Oct. 14
In Lecture:
 Ostrowski's Theorem, concluded
 The topology of local fields
 Approximation Theorems
Associated Reading:
 Borevich and Shafarevitch, see index for "Ostrowski's Theorem".
 Lang, chapter 2.
Video of class (streaming)
Video of class (downloadable)
 Oct. 16
In Lecture:
 Weak and Strong approximation theorems for linear algebraic groups
 Extension fields and absolute values
Associated Reading:

 Platonov and Rapinchuk, sections 1.2, Chapter 2.1, Chapter 7
 Lang, Chapter 2
Video of class (streaming)
Video of class (downloadable)

Oct. 19
In Lecture:
 Oct. 21
In Lecture:
 Approximation Theorems
 Extensions of absolute values
 The Manin Batryev Conjeture
Associated Reading:
Video of class (streaming)
Video of class (downloadable)
 Oct. 23
In Lecture:
 Extensions of absolute values
 Hensel's Lemma
 Selberg's analogy, fields with the same zeta function
Associated Reading:
 Lang, chapter 2
 Sunada's paper on the length spectrum of manifolds
Video of class (streaming)
Video of class (downloadable)

Oct. 26
In Lecture:
 Oct. 30
In Lecture:
 Oct. 28
In Lecture:

Nov. 2
In Lecture:
 Extensions of absolute values (end)
 Hensel's Lemma
Associated Reading:
Video of class (streaming)
Video of class (downloadable)
 Nov. 4
In Lecture:
 Hensel's Lemma and applications
 Dynamical systems and morphisms of varieties
Associated Reading:
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 Nov. 6
In Lecture:
 End of proof of Hensel's Lemma
 Dynamical systems (continued)
Associated Reading:
 Lang, chapters 2 and 5
 Julia Sets
 Galois stochastic processes
 Notes by Larry Shepp on Martingales
Video of class (streaming)
Video of class (downloadable)

Nov. 9
In Lecture:
 Nov. 11
In Lecture:
 Product formula
 Geometry of numbers
 Finiteness of class numbers, finite generation of unit groups
Associated Reading:
Video of class (streaming)
Video of class (downloadable)
 Nov. 13
In Lecture:
 Geometry of numbers (continued)
 Finiteness of class numbers, finite generation of unit groups (continued)
Associated Reading:
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Video of class (downloadable)

Nov. 16
In Lecture:
 Algorithms for computing idea class groups
 Quadratic field examples
Associated Reading:
Video of class (streaming)
Video of class (downloadable)
 Nov. 18
In Lecture:
 Proof of the Minkowski bound
Associated Reading:
Video of class (streaming)
Video of class (downloadable)
 Nov. 20
In Lecture:
 End of the proof of the Minkowski bound, and the function field case
Associated Reading:
Video of class (streaming)
Video of class (downloadable)

Nov. 23
In Lecture:
 The Dirichet Unit theorem, in number fields and function fields
Associated Reading:
Video of class (streaming)
Video of class (downloadable)
 Nov. 25
 Nov. 27

Nov. 30
In Lecture:
 Constructing small generators for discrete groups using fundamental domains
 Applications to generating Sunits
Associated Reading:
Video of class (streaming)
Video of class (downloadable)
 Dec. 2
In Lecture:
 Curves, codes and algebraic integers
 Beginning of class field theory via ray class groups
Associated Reading:
 Here is one Survey article
on codes constructed via algebraic geometry.
 Lang, chapter 10
Video of class (streaming)
Video of class (downloadable)
 Dec. 4
In Lecture:
 Class field theory via ray class groups
 Congruence subgroups of unit groups and the congruence subgroup problem
Associated Reading:
 Lang, chapter 10
 Here is one Survey article
on the congruence subgroup problem.
Video of class (streaming)
Video of class (downloadable)

Dec. 7
In Lecture:
 Frobenius elements
 Class field theory via ray class groups
Associated Reading:
Video of class (streaming)
Video of class (downloadable)
 Dec. 9
No class due to the flu
 Dec. 11
In Lecture:
 Class field theory, continued
Associated Reading:
Video of class (streaming)
Video of class (downloadable)
