Date of Defense
8-4-2026 4:00 PM
Location
Room 2022, F1 Building
Document Type
Thesis Defense
Degree Name
Master of Science in Mathematics
College
College of Science
Department
Mathematical Sciences
First Advisor
Kanat Abdukhalikov
Keywords
Algebraic number fields, ideals, fractional ideals, ideal class groups, trace, norm, discriminant, lattices
Abstract
This thesis investigates algebraic number fields and their rings of integers, which generalize the ring of integers Z in Q. The study focuses on ideals, units, and ideal class groups, which describe the arithmetic structure of number fields and the failure of unique factorization. Key invariants such as the norm, trace, and discriminant are developed and applied, with particular emphasis on quadratic number fields and classical examples such as the Gaussian and Eisenstein integers. Some explicit computations of ideal class groups are carried out. The thesis also explores connections with lattice theory by interpreting rings of integers as lattices and constructing higher-dimensional lattices using Kronecker products.
Included in
Ideals and Lattices in Number Fields
Room 2022, F1 Building
This thesis investigates algebraic number fields and their rings of integers, which generalize the ring of integers Z in Q. The study focuses on ideals, units, and ideal class groups, which describe the arithmetic structure of number fields and the failure of unique factorization. Key invariants such as the norm, trace, and discriminant are developed and applied, with particular emphasis on quadratic number fields and classical examples such as the Gaussian and Eisenstein integers. Some explicit computations of ideal class groups are carried out. The thesis also explores connections with lattice theory by interpreting rings of integers as lattices and constructing higher-dimensional lattices using Kronecker products.