The Distribution of Class Numbers in a Family of Real Quadratic Fields by Levi Borevitz '23

The Distribution of Class Numbers in a Family of Real Quadratic Fields by Levi Borevitz ’23, Mathematics Senior Thesis Defense, Monday, May 15, 1 – 1:40 pm, North Science Building 113, Wachenheim.

Abstract:  Class numbers are a fundamental object of study in number theory.  Originally introduced by Gauss in his exploration of quadratic forms, they play an important role in algebraic number theory (measuring the failure of unique factorization in the ring of integers of a number field), analytic number theory (appearing in an explicit formula for the value of Dirichlet L-functions), and elementary number theory (appearing in a formula for a linear combination of the decimal digits of 1/n). Despite over two centuries of intense research, many basic questions about class numbers remain unanswered, for example whether there exist infinitely many real quadratic fields of class number 1 (a conjecture due to Gauss).  In this talk, we introduce class numbers, as well as some other number field invariants like Dedekind L-functions, the discriminant, and the regulator. Then we discuss results related to the average number of number fields with a given class number, including new results from the speaker’s undergraduate thesis.