Dedekind–Kummer theorem

(Redirected from 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.[1]

Statement for number fields

edit

Let   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

edit

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 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
  1. ^ a b Neukirch, Jürgen (1999). Algebraic number theory. Berlin: Springer. pp. 48–49. ISBN 3-540-65399-6. OCLC 41039802.
  2. ^ Conrad, Keith. "FACTORING AFTER DEDEKIND" (PDF).