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

### Title

Topics in Algebraic Number Theory.

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### Description

Current topics of research interest.

## 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.

### 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.

### Minimal learning outcomes

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. 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. 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. 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. 5. Arithmetic of cyclotomic fields, the Kronecker-Weber Theorem. 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.

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
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
3. Ramification Theory:
• Relative extensions
• Relative discriminant and Dedekind's criterion for ramification in terms of discriminant
• Higher ramification groups
• Hilbert theory of ramification
4. Splitting of Primes:
• Frobenius map
• Artin symbol
• Artin map
• Splitting of primes in Abelian extensions in terms of Artin map
• Rudimentary class field theory
• Examples - quadratic and cyclotomic extensions
5. Arithmetic of cyclotomic fields, and the Kronecker-Weber Theorem
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

In addition, time permitting, the instructor may want to add to the list topics of special to him/her.

### Textbooks

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