Date of Completion
Spring 5-1-2024
Thesis Advisor(s)
Keith Conrad
Honors Major
Mathematics
Disciplines
Number Theory
Abstract
A Hilbert symbol has the value 1 or −1 depending on the existence of solutions to a certain quadratic equation in a local field, R, or C. Hilbert reciprocity states that for a number field F and two nonzero a and b in F, the product of Hilbert symbols associated to a and b at all the places of F is 1. That is, these Hilbert symbols are −1 for a finite, even number of places of F . Hilbert reciprocity when F = Q is equivalent to the classical quadratic reciprocity law, so Hilbert reciprocity in number fields can be considered an extension of quadratic reciprocity to all number fields. This thesis presents a proof of Hilbert reciprocity in all number fields and then uses it to derive a quadratic reciprocity law in three quadratic rings that resembles the quadratic reciprocity law in Z, both in terms of Legendre symbols and Jacobi symbols.
Recommended Citation
Snyder, Dillon, "Hilbert Reciprocity over Number Fields" (2024). Honors Scholar Theses. 984.
https://digitalcommons.lib.uconn.edu/srhonors_theses/984