Math 686R: Topics in Algebraic Number Theory.
Topics in Algebraic Number Theory.
Math 372 and permission of the Instructor. In general, the prerequisites will depend on the material covered.
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.
The pre-requisites in the Catalog are adequate. Depending on the instructor and with her/his permission, a sound understanding of basic concepts of complex analysis, abstract algebra and number theory might be adequate, perhaps at the level of Math 352, 371, 372 and 487.
Minimal learning outcomes
- Generalization of the Unique Factorization Theorem from rationals to number fields; Basic definitions:
- Number fields
- Algebraic integers in a number field
- Integral bases
- 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
- 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
- Ramification Theory:
- Relative extensions
- Relative discriminant and Dedekind's criterion for ramification in terms of discriminant
- Higher ramification groups
- Hilbert theory of ramification
- 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
- Arithmetic of cyclotomic fields, and the Kronecker-Weber Theorem
- 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 interest to him/her.
Possible textbooks for this course include, (but are not limited to):
- D. Marcus, Number Fields (Universitext)