1. Jason Holman Mathematics Number Theory Algebraic Theory Global Fields for Complex Multiplication Local and “p”-adic fields

2. 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 Things To Remember

3. 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 What is Algebraic Number Theory?

4. 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 Topics That Would Be Studied In a Class on Algebraic Number Theory

5. 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 studied in Algebraic Number Theory?

6. Integer Factorization December 2003 $10000 challenge What is this used for?

7. Primality Test Pell’s Equations Diophantine Equations Riemann Hypothesis What is it used for?

8. Pell’s Equations http://mathworld.wolfram.com/PellEquation.html Diophantine Equations http://mathworld.wolfram.com/DiophantineEquation2nd Powers.html http://mathworld.wolfram.com/DiophantineEquation2nd Powers.html Riemann Hypothesis http://mathworld.wolfram.com/RiemannHypothesis.html For More Info

9. 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 Recap

10. Stein, W. (May, 2005). Introduction to algebraic number theory. Unpublished paper. http://mathworld.wolfram.com/Field.html Sources