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
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
.
[2] 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
akF=\{a\inlO\midalO\subseteqlo[\theta]\}
(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
akPi=akplO+(fi(\theta)lO)
and the
are the polynomials
lifted to
.
References
- Book: Neukirch, Jürgen. Algebraic number theory. 1999. Springer. 3-540-65399-6. Berlin. 48–49. 41039802.
- Web site: Conrad. Keith. FACTORING AFTER DEDEKIND.