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. Statement for number fields 
Let  be a number field such that  for  and let  be the minimal polynomial for  over . For any prime  not dividing , writewhere  are monic irreducible polynomials in . Then  factors into prime ideals assuch that . 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  (i.e. their product is ). 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 . References 

Algebraic number theory