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

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)× = and (ℤ₂e)× = × <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.