Monday 
Wednesday 
Friday 
Sept. 3
In Lecture:
 Sept. 5
In Lecture:
 Sept. 7
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 (streaming)
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Sept. 10
In Lecture:
 Norms and Traces
 Integral 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 (streaming)
Video of class (downloadable)
 Sept. 12
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 (streaming)
Video of class (downloadable)
 Sept. 14
In Lecture:
No class  next class will be Monday, Sept. 17.

Sept. 17
In Lecture:
Associated Reading:
 Lang, Chapter I, sections 1.2, 1.3, 1.4, 1.5.
 Samuel, chapter 2.
Video of class (streaming)
Video of class (downloadable)
 Sept. 19
In Lecture:
 Discriminants, continued
 Unramified extensions of rings
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 (streaming)
Video of class (downloadable)
 Sept. 21
In Lecture:
No class  next class will be Friday, Sept. 28

Sept. 24
In Lecture:
No class  next class will be Friday, Sept. 28
 Sept. 26
In Lecture:
No class  next class will be Friday, Sept. 28
 Sept. 28
In Lecture:
 Rings of cyclotomic integers
 Statement of Kronecker Weber Theorem, Hilbert's 12th problem
Associated Reading:
 Samuel, chapter 2, section 2.9
 Samuel, chapter 3.
 Lang, chapter 4, section 4.1.
 Lang, chapter 1, section 1.6
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Oct. 1
In Lecture:
Associated Reading:
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 Oct. 3
In Lecture:
 Constructing abelian extensions of imaginary quadratic fields via elliptic functions.
 Definition of Dedekind rings
 Integral closures of Dedekind rings in finite separable extensions of their fraction fields
Associated Reading:
 The book Elliptic functions and rings of integers by Ph. CassouNogues and M. J. Taylor.
 Washington, chapters 1 and 2.
 Samuel, chapter 2 section 2.9 and 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 (steaming)
Video of class (downloadable)
 Oct. 5
In Lecture:
 Fractional ideals of Dedekind rings form a group
 Ideal class groups, Dedekind subrings of function fields of curves.
 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 (steaming)
Video of class (downloadable)

Oct. 8
In Lecture:
 Completion of the proof that the fractional ideals of a Dedekind ring form a group
Associated Reading:
 Samuel, chapter 5, sections 5.1 and 5.2.
 Lang, chapter 1, section 1.6, 1.7, 1.8
Video of class (streaming)
Video of class (downloadable)
 Oct. 10
In Lecture:
 Decomposition of prime ideals in finite separable extensions
Associated Reading:
 Samuel, chapter 5, sections 5.1 and 5.2.
 Lang, chapter 1, section 1.6, 1.7, 1.8
Video of class (steaming)
Video of class (downloadable)
 Oct. 12
In Lecture:
 Decomposition of prime ideals in Galois extensions
 Decomposition and Inertia groups
Associated Reading:
 Samuel, chapter 5, sections 5.1 and 5.2.
 Lang, chapter 1, section 1.6, 1.7, 1.8
Video of class (steaming)
Video of class (downloadable)

Oct. 15
In Lecture:
 Decomposition of primes in extensions generated by a single element
 Valuations and discrete valuations.
 Examples of discrete valuation rings
 Discrete valuations of fraction fields of Dedekind rings
Associated Reading:
Video of class (steaming)
Video of class (downloadable)
 Oct. 17
In Lecture:
 Discrete valuations on higher dimensional function fields, connection with blowups
 Absolute values on fields and completions of fields
 The canonical absolute values of number fields and function fields
 The product formula
 Formal power series expressions for elements of completions.
Associated Reading:
Video of class (steaming)
Video of class (downloadable)
 Oct. 19
In Lecture:
 More on formal power series for elements of completions
 Pictures of Z_p
 When Iadic completions of rings are compact and totally disconnected.
Associated Reading:
Video of class (streaming)
Video of class (downloadable )

Oct. 22
In Lecture:
 Oct. 24
In Lecture:
 Adeles of global fields
 Weak and Strong approximation theorems
Associated Reading:
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Video of class (downloadable)
 Oct. 26
In Lecture:

Oct. 29
In Lecture:
 No class  Hurricaine Sandy!
 Oct. 31
In Lecture:
 Minkowski's Lemma
 End of the proof of the strong approximation theorem for rings of integers
Associated Reading:
Video of class (streaming)
Video of class (downloadable)
 Nov. 2
In Lecture:

Nov. 5
In Lecture:
 Nov. 7
In Lecture:
 Quasicrystals and the strong approximation theorem
 Weak and strong approximation theorem for algebraic groups.
Associated Reading:
Video of class (streaming)
Video of class (downloadable)
 Nov. 9
In Lecture:
 The Riemann Roch theorem on curves, and strong approximation
Associated Reading:
 Hartshorne, "Algebraic Geometry", Chapter IV.
 Lang, chapter 2
Video of class (streaming)
Video of class (downloadable)

Nov. 12
In Lecture:
 The Riemann Roch theorem on curves, continued
 Finiteness of Pic^0 for curves over finite fields.
Associated Reading:
 Hartshorne, "Algebraic Geometry", Chapter IV.
 Lang, chapter 2
Video of class (streaming)
Video of class (downloadable)
 Nov. 14
In Lecture:
 Riemann Roch and Goppa codes
 Geometry of numbers
 Finiteness of class numbers, finite generation of unit groups
Associated Reading:
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Video of class (downloadable)
 Nov. 16
No Class!

Nov. 19
No Class!
 Nov. 21
No Class!
 Nov. 23
No Class!  Thanksgiving!

Nov. 26
In Lecture:
 Finiteness of ideal class groups (classical proof)
Associated Reading:
Video of class (streaming)
Video of class (downloadable)
 Nov. 28
In Lecture:
 Finiteness of class numbers (classical proof, completed)
 Dirichlet unit theorem (classical proof)
Associated Reading:
Video of class (streaming)
Video of class (downloadable)
 Nov. 30
In Lecture:
 Dirichlet unit theorem (classical proof, completed)
Associated Reading:
Video of class (streaming  to be posted)
Video of class (downloadable)

Dec. 3
In Lecture:
 Lenstra's approach to the unit theorem.
 Examples of computing units
 Continued fractions and real quadratic units
 Arakelov theory in dimension one
Associated Reading:
Video of class (streaming)
Video of class (downloadable)
 Dec. 5
In Lecture:
 Arakelov theory in dimension 1, continued
Associated Reading:
 Lang, chapters 2 and 5
 The paper "Presentation de la Theorie d'Arakelov", by L. Szpiro, A.M.S. Contemporary Mathematics series no. 67.
Video of class (streaming)
Video of class (downloadable)
 Dec. 7
In Lecture:
 End of Arakelov theory in dimension 1.
Associated Reading:
 Lang, chapters 2 and 5
 The paper "Presentation de la Theorie d'Arakelov", by L. Szpiro, A.M.S. Contemporary Mathematics series no. 67.
Video of class (streaming)
Video of class (downloadable)
