Dedekind–Kummer theorem explained

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

Let

K be a number field such that K = \Q(\alpha) for \alpha \in l O _{K} and let f be the minimal polynomial for \alpha over \Z[x] . For any prime p not dividing [ l O _{K} : \Z[\alpha]] , write$$f(x)\; \backslash equiv\; \backslash pi\_1\; (x)^\; \backslash cdots\; \backslash pi\_g(x)^\; \backslash mod\; p$$where \pi _{i} (x) are monic irreducible polynomials in F _{p[x]} . Then (p) = p l O _{K} factors into prime ideals as$$(p)\; =\; \backslash mathfrak\; p\_1^\; \backslash cdots\; \backslash mathfrak\; p\_g^$$ such that N( ak p _{i)} = p ^{\deg \pi _{i}} .^[2]

Statement for Dedekind Domains

The Dedekind-Kummer theorem holds more generally than in the situation of number fields: Let

l o be a Dedekind domain contained in its quotient field K , L/K a finite, separable field extension with L = K[\theta] for a suitable generator \theta and l O the integral closure of l o . The above situation is just a special case as one can choose l o = \Z, K = \Q, l O = l O _{L} ).

If

( 0 ) ≠ ak p\subseteq l o is a prime ideal coprime to the conductor ak F = \{a\in l O\mid a l O\subseteq l o[\theta]\} (i.e. their sum is l O ). Consider the minimal polynomial f\in l o[x] of \theta . The polynomial \overline f\in( l o / ak p)[x] has the decomposition $$\backslash overline\; f=\backslash overline^\backslash cdots\; \backslash overline^$$ with pairwise distinct irreducible polynomials \overline{f _{i}} .The factorization of ak p into prime ideals over l O is then given by $$\backslash mathfrak\; p=\backslash mathfrak\; P\_1^\backslash cdots\; \backslash mathfrak\; P\_r^$$ where ak P _{i=akplO+(f} _{i(\theta)lO)} and the f _{i} are the polynomials \overline{f _{i}} lifted to l o[x] .

References

1. Book: Neukirch, Jürgen. Algebraic number theory. 1999. Springer. 3-540-65399-6. Berlin. 48–49. 41039802.
2. Web site: Conrad. Keith. FACTORING AFTER DEDEKIND.