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\inlOK

and let

f

be the minimal polynomial for

\alpha

over

\Z[x]

. For any prime

p

not dividing

[lOK:\Z[\alpha]]

, writef(x) \equiv \pi_1 (x)^ \cdots \pi_g(x)^ \mod pwhere

\pii(x)

are monic irreducible polynomials in

Fp[x]

. Then

(p)=plOK

factors into prime ideals as(p) = \mathfrak p_1^ \cdots \mathfrak p_g^ such that

N(akpi)=

\deg\pii
p
.[2]

Statement for Dedekind Domains

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

lo

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

lO

the integral closure of

lo

. The above situation is just a special case as one can choose

lo=\Z,K=\Q,lO=lOL

).

If

(0)akp\subseteqlo

is a prime ideal coprime to the conductor

akF=\{a\inlO\midalO\subseteqlo[\theta]\}

(i.e. their sum is

lO

). Consider the minimal polynomial

f\inlo[x]

of

\theta

. The polynomial

\overlinef\in(lo/akp)[x]

has the decomposition \overline f=\overline^\cdots \overline^ with pairwise distinct irreducible polynomials

\overline{fi}

.The factorization of

akp

into prime ideals over

lO

is then given by \mathfrak p=\mathfrak P_1^\cdots \mathfrak P_r^ where

akPi=akplO+(fi(\theta)lO)

and the

fi

are the polynomials

\overline{fi}

lifted to

lo[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.