Math 702 schedule
| 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)
Video of class (downloadable)
|
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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Video of class (downloadable)
|
Oct. 1
In Lecture:
Associated Reading:
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Video of class (downloadable)
| 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. Cassou-Nogues 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 blow-ups
- 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 I-adic 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:
Video of class (streaming)
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:
- Quasi-crystals 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:
Video of class (streaming)
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:
- A paper by Hendrik Lenstra on number theoretic algorithms in the Bull. of the A. M. S.
- Pell's equation and continued fractions
- 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. 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)
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