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] It is named after Richard Dedekind who developed the theorem based on the work of Ernst Kummer.
[citation needed][when?]
be a number field such that
( α )
α ∈
be the minimal polynomial for
α
For any prime
[ α ] ]
, write
π
π
π
are monic irreducible polynomials in
factors into prime ideals as
π
is the ideal norm.
[2] 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
the integral closure of
The above situation is just a special case as one can choose
is a prime ideal coprime to the conductor
Consider the minimal polynomial
The polynomial
with pairwise distinct irreducible polynomials
into prime ideals over
are the polynomials