In algebraic number theory, the Dedekind–Kummer theorem describes how a prime ideal in a Dedekind domain factors over the domain's integral closure.[1]
Statement for number fields
editLet be a number field such that for and let be the minimal polynomial for over . For any prime not dividing , write where are monic irreducible polynomials in . Then factors into prime ideals as such that , where is the ideal norm.[2]
Statement for Dedekind Domains
editThe 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 sum 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 .[1]
References
edit- ^ a b Neukirch, Jürgen (1999). Algebraic number theory. Berlin: Springer. pp. 48–49. ISBN 3-540-65399-6. OCLC 41039802.
- ^ Conrad, Keith. "FACTORING AFTER DEDEKIND" (PDF).