Math 620 schedule

Monday Wednesday Friday
Sept. 7

In Lecture:
  • No Class!
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

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Sept. 11

In Lecture:
  • No Class!
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.

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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.

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Sept. 18

In Lecture:
  • No class!
Sept. 21

In Lecture:
  • Discriminants

Associated Reading:
  • Samuel, chapter 2, sections 2.7, 2.8.
  • Lang, chapter 3, section 3.3.

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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:
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Sept. 25

In Lecture:
  • No Class!
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.

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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.

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Video of class (downloadable)
Oct. 2

In Lecture:
  • No Class!
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:
  • Lang, chapter 2.

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Oct. 9

In Lecture:
  • The time and location of the class will be sent by e-mail the night of Oct. 8

Associated Reading:
  • Absolute values and completions, continued

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Video of class (downloadable)
Oct. 12

In Lecture:
  • Ostrowski's Theorem

Associated Reading:
  • Borevich and Shafarevitch, see index for "Ostrowski's Theorem".
  • Lang, chapter 2.

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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.

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Video of class (downloadable)
Oct. 16

In Lecture:
  • Weak and Strong approximation theorems for linear algebraic groups
  • Extension fields and absolute values

Associated Reading:
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Video of class (downloadable)
Oct. 19

In Lecture:
  • No class - fall break!
Oct. 21

In Lecture:
  • Approximation Theorems
  • Extensions of absolute values
  • The Manin Batryev Conjeture

Associated Reading:
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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:
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Video of class (downloadable)
Oct. 26

In Lecture:
  • No class!
Oct. 30

In Lecture:
  • No class!
Oct. 28

In Lecture:
  • No class!
Nov. 2

In Lecture:
  • Extensions of absolute values (end)
  • Hensel's Lemma

Associated Reading:
  • Lang, chapter 2

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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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Video of class (downloadable)
Nov. 6

In Lecture:
  • End of proof of Hensel's Lemma
  • Dynamical systems (continued)

Associated Reading:
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Video of class (downloadable)
Nov. 9

In Lecture:
  • No class!
Nov. 11

In Lecture:
  • Product formula
  • Geometry of numbers
  • Finiteness of class numbers, finite generation of unit groups

Associated Reading:
  • Lang, chapters 2 and 5

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Video of class (downloadable)
Nov. 13

In Lecture:
  • Geometry of numbers (continued)
  • Finiteness of class numbers, finite generation of unit groups (continued)

Associated Reading:
  • Lang, chapters 2 and 5

Video of class (streaming)
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:
  • Lang, chapters 2 and 5

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Video of class (downloadable)
Nov. 20

In Lecture:
  • End of the proof of the Minkowski bound, and the function field case

Associated Reading:
  • Lang, chapters 2 and 5

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Video of class (downloadable)
Nov. 23

In Lecture:
  • The Dirichet Unit theorem, in number fields and function fields

Associated Reading:
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Video of class (downloadable)
Nov. 25

    No class!
Nov. 27

    No class!
Nov. 30

In Lecture:
  • Constructing small generators for discrete groups using fundamental domains
  • Applications to generating S-units

Associated Reading:
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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:
  • Lang, chapter 10

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:
  • Lang, chapter 10

Video of class (streaming)
Video of class (downloadable)

Last updated: 12/11/09
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