Structure of unit groups of quotient rings of integers in Some cubic fields

Abstract

In the ring of integer ℤ, the structure of unit groups of quotient rings, denoted by (ℤn)×, is known. By the Chinese remainder theorem, the study of structure of (ℤn)× is reduced to study the structure of (ℤpe)× for all primes p and natural numbers e. It is well known that for an odd prime p, (ℤpe)× is a cyclic group of order ϕ(pe) for all natural number e, while (ℤ₂)× = {1}, (ℤ₄)× = <−1> and (ℤ₂e)× = <−1> × <5> for all natural numbers e ≥ 3. For a number field K, denote the ring of integers in K by OK. In this thesis we will study the structure of unit groups of quotient rings, denoted by (OK/A)×, for any ideal A of OK for cubic fields K with square-free discriminant.