Transcript Algebraic Number Theory

Jason Holman
• Mathematics
• Number Theory
• Algebraic Theory
• Global Fields for Complex Multiplication
• Local and “p”-adic fields
Things To Remember
Review of some definitions
•Field
• A field is any set of elements that satisfies the field
axioms (associative, commutative, distributive,
inverse, identity) for both addition and
multiplication and is a commutative division algebra
• Other Useful information
• Finite group theory
• Commutative rings and quotient rings
• Elementary number theory
What is Algebraic Number Theory?
Defined as…
• A number field K is a finite algebraic extension of the
rational numbers
What it involves…
• Using techniques from algebra and finite group theory to
gain a deeper understanding of fields
Topics That Would Be Studied In a
Class on Algebraic Number Theory
• Rings of integers of number fields
• Unique factorization of ideals in Dedekind
domains
• Structure of the group of units of the ring of
integers
• Discriminant and different
• Quadratic and biquadratic fields
• Several others
What is studied in Algebraic
Number Theory?
Algebraic number theory has roots in
several areas including…
• Fields
• Rings of integers of number fields
• Unit groups
• Ideal class groups
• Norms
• Traces
• Many others
What is this used for?
Integer Factorization
• December 2003 $10000 challenge
What is it used for?
• Primality Test
• Pell’s Equations
• Diophantine Equations
• Riemann Hypothesis
For More Info
Pell’s Equations
• http://mathworld.wolfram.com/PellEquation.html
Diophantine Equations
• http://mathworld.wolfram.com/DiophantineEquation2nd
Powers.html
Riemann Hypothesis
• http://mathworld.wolfram.com/RiemannHypothesis.html
Recap
• Algebraic Number Theory is a basis for
several other, deeper, areas of math
• Some of these areas include fields, rings,
and groups
• It’s main uses occur in integer
factorization, primality tests, and Pell’s
equations
Sources
Stein, W. (May, 2005). Introduction to
algebraic number theory. Unpublished
paper.
http://mathworld.wolfram.com/Field.html