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

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=== Prerequisite ===
 
=== Prerequisite ===
[[Math 352]], [[Math 487|487]], [[Math 671|671]], [[Math 672|672]].
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Math 372 and permission of the Instructor.  In general, the prerequisites will depend on the material covered.
  
 
=== Description ===
 
=== Description ===
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== Desired Learning Outcomes ==
 
== Desired Learning Outcomes ==
To gain familiarity with working in general settings, e.g.  Abelian varieties over number fields, not just of elliptic curves over rationals.
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Gaining mastery over an advanced area of algebraic number theory of interest in research.
 +
 
 
=== Prerequisites ===
 
=== Prerequisites ===
A sound understanding of basic concepts of complex analysis, abstract algebra and number theory, at the level of Math 352, 371, 372 and 487 may suffice.
+
Math 372 and permission of the Instructor.  In general, the prerequisites will depend on the material covered.
 +
 
 
=== Minimal learning outcomes ===
 
=== Minimal learning outcomes ===
 
+
These cannot be specified uniquely for a topics course. Past topics have included "Primes of the form <i>x<sup>2</sup>+Ny<sup>2</sup></i>," "The Proof of Fermat's Last Theorem," and "Computational Algebraic Number Theory." 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. Generalization of the Unique Factorization Theorem from rationals to number fields: Basic Definitions - number fields, algebraic integers in a number field, integral bases, discriminant, norms of ideals, finiteness of ideals of bounded norm, class number, finiteness of class number, Dedekind's Unique Factorization Theorem for ideals of a number field.
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2. Geometry of numbers: Minkowski's lemma on lattice points, Logarithmic spaces, Dirichlet's Unit Theorem for the units of the ring of integers of a number field, theorems of Minkowski and of Hermite on discriminants of number fields.
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3. Ramification Theory: Relative extensions, relative discriminant and Dedekind's criterion for ramification in terms of discriminant, higher ramification groups and Hilbert theory of ramification.
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4  Splitting of Primes: Frobenius map, Artin symbol, Artin map and splitting of primes in Abelian extensions in terms of Artin map, rudimentary class field theory, Examples - quadratic and cyclotomic extensions.
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5. Arithmetic of cyclotomic fields, the Kronecker-Weber Theorem.
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6. Dedekind zeta function, 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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<div style="-moz-column-count:2; column-count:2;">
 
<div style="-moz-column-count:2; column-count:2;">
#Generalization of the Unique Factorization Theorem from rationals to number fields; Basic definitions:
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# Primes of the form <i>x<sup>2</sup>+Ny<sup>2</sup></i>
#* 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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</div>
 
</div>
 
In addition, time permitting, the instructor may want to add to the list topics of special to him/her.
 
  
 
=== Textbooks ===
 
=== Textbooks ===
  
Possible textbooks for this course include (but are not limited to):
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Possible textbooks for this course include, (but are not limited to):  
 
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* D. Cox, Primes of the form x^2+Ny^2
*
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* H. Cohen, A course in computational number theory
  
 
=== Additional topics ===
 
=== Additional topics ===
 +
None
  
 
=== Courses for which this course is prerequisite ===
 
=== Courses for which this course is prerequisite ===
 +
 +
None
  
 
[[Category:Courses|686]]
 
[[Category:Courses|686]]

Latest revision as of 15:20, 11 July 2016

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

Gaining mastery over an advanced area of algebraic number theory of interest in research.

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. Past topics have included "Primes of the form x2+Ny2," "The Proof of Fermat's Last Theorem," and "Computational Algebraic Number Theory." 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 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
    • 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
    • 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

None

Courses for which this course is prerequisite

None