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Transcript
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 Unpublished paper.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
