Dr. Olivia Beckwith, University of Illinois Urbana Urbana-Champaign
Title: Divisibility properties of class numbers of quadratic fields
Abstract: Gauss was the first to count classes of binary quadratic forms up to matrix equivalence, and the number of equivalence classes for a fixed discriminant, the class number, measures the obstruction to unique factorization into primes for quadratic number fields. Information about class numbers percolates into many branches of number theory, including the theory of L-functions via Dirichlet's class number formula, and elliptic curves via the work of Birch and Swinnerton-Dyer. This colloquium-style talk will begin with a brief introduction to algebraic number theory and a review of the definition and significance of the class number of quadratic fields, as well as some of the important results in the history of their study. Then I will discuss some of my work in this area, concerning the divisibility properties of class numbers.