Dedekind–Kummer theorem
In algebraic number theory, the Dedekind–Kummer theorem describes how a prime ideal in a Dedekind domain factors over the domain's integral closure. It is named after Richard Dedekind who developed the theorem based on the work of Ernst Kummer.
Statement for number fields
Let be a number field and the ring of algebraic integers in. Let and be the minimal polynomial of over. For any prime not dividing the index, writewhere are monic irreducible polynomials in. Then, the ideal factors into prime ideals as
such that, where is the ideal norm.
Statement for Dedekind domains
The Dedekind-Kummer theorem holds more generally than in the situation of number fields: Let be a Dedekind domain contained in its quotient field, a finite, separable field extension with for a suitable generator and the integral closure of. The above situation is just a special case as one can choose ).If is a prime ideal coprime to the conductor . Consider the minimal polynomial of. The polynomial has the decomposition
with pairwise distinct irreducible polynomials.
The factorization of into prime ideals over is then given by where and the are the polynomials lifted to.