Difference between revisions of "Math 686R: Topics in Algebraic Number Theory."

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#Generalization of the Unique Factorization Theorem from rationals to number fields; Basic definitions:
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# Primes of the form <it>x<sup>2</sup>+Ny<sup>2</sup>
#* Number fields
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#* Classical Topics
#* Algebraic integers in a number field
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#* Equivalence of binary quadratic forms
#* Integral bases
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#* Composition of Binary quadratic forms
#* Discriminant
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#* Relation between binary quadratic forms and ideal class groups of quadratic fields
#* Norms of ideals
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#* Representations of integers by binary quadratic forms
#* Finiteness of ideals of bounded norm
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# Class field theory
#* Class number
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#* Artin Reciprocity and the Artin Map
#* Finiteness of class number
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#* The Existence and Conductor Theorems
#* Dedekind's Unique Factorization Theorem for ideals of a number field
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#* Cebotarev Density Theorem
#Geometry of numbers:
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#* Identifying Ray Class Fields and Ring Class Fields
#* Minkowski's lemma on lattice points
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# Complex Analytic Techniques
#* Logarithmic spaces
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#* Elliptic Functions
#* Dirichlet's Unit Theorem for the units of the ring of integers of a number field
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#* Properties of the Weierstrass p-function
#* Theorems of Minkowski and of Hermite on discriminants of number fields
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#* Invariants of Lattices
#Ramification Theory:
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#* Modular forms--definitions and properties
#* Relative extensions
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#* Singular values of modular forms
#* Relative discriminant and Dedekind's criterion for ramification in terms of discriminant
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#* The modular equation
#* Higher ramification groups
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#* Computing generators for ring class fields
#* Hilbert theory of ramification
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#Splitting of Primes:
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#* Frobenius map
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#* Artin symbol
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#* Artin map
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#* Splitting of primes in Abelian extensions in terms of Artin map
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#* Rudimentary class field theory
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#* Examples - quadratic and cyclotomic extensions
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#Arithmetic of cyclotomic fields, and the Kronecker-Weber Theorem
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#Dedekind zeta function
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#* The class number formula - the formula which relates the residue of the Dedekind zeta function of a number field at s=1 to its class number, regulator and discriminant
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Possible textbooks for this course include, (but are not limited to):  
 
Possible textbooks for this course include, (but are not limited to):  
* D. Marcus, Number Fields (Universitext)
 
 
* D. Cox, Primes of the form x^2+Ny^2
 
* D. Cox, Primes of the form x^2+Ny^2
 
* H. Cohen, A course in computational number theory
 
* H. Cohen, A course in computational number theory

Revision as of 08:14, 16 March 2015

Catalog Information

Title

Topics in Algebraic Number Theory.

Credit Hours

3

Prerequisite

Math 372 and permission of the Instructor. In general, the prerequisites will depend on the material covered.

Description

Current topics of research interest.

Desired Learning Outcomes

To gain familiarity with working in general settings, e.g. elliptic curves over number fields, and not just over rationals.

Prerequisites

Math 372 and permission of the Instructor. In general, the prerequisites will depend on the material covered.

Minimal learning outcomes

These cannot be specified uniquely for a topics course. The following example gives a clear indication of the level of difficulty appropriate to the course. Students should know the technical terms, and be able to implement the methods taught in the course to work associated problems, including proving results of suitable accessibility.

  1. Primes of the form <it>x2+Ny2
    • Classical Topics
    • Equivalence of binary quadratic forms
    • Composition of Binary quadratic forms
    • Relation between binary quadratic forms and ideal class groups of quadratic fields
    • Representations of integers by binary quadratic forms
  2. Class field theory
    • Artin Reciprocity and the Artin Map
    • The Existence and Conductor Theorems
    • Cebotarev Density Theorem
    • Identifying Ray Class Fields and Ring Class Fields
  3. Complex Analytic Techniques
    • Elliptic Functions
    • Properties of the Weierstrass p-function
    • Invariants of Lattices
    • Modular forms--definitions and properties
    • Singular values of modular forms
    • The modular equation
    • Computing generators for ring class fields

Textbooks

Possible textbooks for this course include, (but are not limited to):

  • D. Cox, Primes of the form x^2+Ny^2
  • H. Cohen, A course in computational number theory

Additional topics

Courses for which this course is prerequisite