Factor rings of the ring of integers in a number field

Question

The factor rings of the ordinary integers $\mathbb Z$ are the well-known residual classes $\mathbb Z_n$. For the Gaussian integers $\mathbb Z[i]$ the factor rings are studied in

1) J. T. Cross, The Euler ϕ-function in the Gaussian integers, Amer. Math. Monthly 90 (1983) 518–528.

2) Dresden, Greg(1-WLEE); Dymàček, Wayne M.(1-WLEE) Finding factors of factor rings over the Gaussian integers. Amer. Math. Monthly 112 (2005), no. 7, 602–611.

There are articles on the factor rings of the ring of integers of an algebraic number field?

Answer 1

The determination of the structure of the multiplicative group $({\mathfrak o}/{\mathfrak p}^n)^\times$, which could not be done with Dedekind's theory of ideals, was one of the first successes of Hensel's $\mathfrak p$-adic numbers. You can read all about it in Hasse's Number Theory, Chapter 15, completed by Nakagoshi (Norikata), The structure of the multiplicative group of residue classes modulo ${\mathfrak p}^{N+1}$. Nagoya Math. J. 73 (1979), 41–60, available at Link.

Answer 2

An ideal $I$ in the ring of integers $\mathcal O$ of some number field factors into a product $I = P_1^{e_1} \cdots P_r^{e_r}$ of prime ideals. Hence, $$ \mathcal O/I \cong \prod \mathcal O/P_i^{e_i}. $$ So the question becomes: What is the structure of $\mathcal O/P^e$ where $P$ is a prime ideal?

The additive and multiplicative structures of these rings are described in the following survey papers.

Elia, Interlando, Rosenbaum, On the structure of residue rings of prime ideals in algebraic number fields Part I: unramified primes. Int. Math. Forum 5 (2010), no. 53-56, 2795–2808.

Elia, Interlando, Rosenbaum, On the structure of residue rings of prime ideals in algebraic number fields—Part II: ramified primes. Int. Math. Forum 6 (2011), no. 9-12, 565–589.