Theorem in algebraic number theory
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
{\displaystyle K}
be a number field such that
K
=
Q
(
α
)
{\displaystyle K=\mathbb {Q} (\alpha )}
for
α
∈
O
K
{\displaystyle \alpha \in {\mathcal {O}}_{K}}
and let
f
{\displaystyle f}
be the minimal polynomial for
α
{\displaystyle \alpha }
over
Z
[
x
]
{\displaystyle \mathbb {Z} [x]}
. For any prime
p
{\displaystyle p}
not dividing
[
O
K
:
Z
[
α
]
]
{\displaystyle [{\mathcal {O}}_{K}:\mathbb {Z} [\alpha ]]}
, write
f
(
x
)
≡
π
1
(
x
)
e
1
⋯
π
g
(
x
)
e
g
mod
p
{\displaystyle f(x)\equiv \pi _{1}(x)^{e_{1}}\cdots \pi _{g}(x)^{e_{g}}\mod p}
where
π
i
(
x
)
{\displaystyle \pi _{i}(x)}
are monic
irreducible polynomials in
F
p
[
x
]
{\displaystyle \mathbb {F} _{p}[x]}
. Then
(
p
)
=
p
O
K
{\displaystyle (p)=p{\mathcal {O}}_{K}}
factors into prime ideals as
(
p
)
=
p
1
e
1
⋯
p
g
e
g
{\displaystyle (p)={\mathfrak {p}}_{1}^{e_{1}}\cdots {\mathfrak {p}}_{g}^{e_{g}}}
such that
N
(
p
i
)
=
p
deg
π
i
{\displaystyle N({\mathfrak {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
o
{\displaystyle {\mathcal {o}}}
be a Dedekind domain contained in its quotient field
K
{\displaystyle K}
,
L
/
K
{\displaystyle L/K}
a finite, separable field extension with
L
=
K
[
θ
]
{\displaystyle L=K[\theta ]}
for a suitable generator
θ
{\displaystyle \theta }
and
O
{\displaystyle {\mathcal {O}}}
the integral closure of
o
{\displaystyle {\mathcal {o}}}
. The above situation is just a special case as one can choose
o
=
Z
,
K
=
Q
,
O
=
O
L
{\displaystyle {\mathcal {o}}=\mathbb {Z} ,K=\mathbb {Q} ,{\mathcal {O}}={\mathcal {O}}_{L}}
).
If
(
0
)
≠
p
⊆
o
{\displaystyle (0)\neq {\mathfrak {p}}\subseteq {\mathcal {o}}}
is a prime ideal coprime to the conductor
F
=
{
a
∈
O
∣
a
O
⊆
o
[
θ
]
}
{\displaystyle {\mathfrak {F}}=\{a\in {\mathcal {O}}\mid a{\mathcal {O}}\subseteq {\mathcal {o}}[\theta ]\}}
(i.e. their sum is
O
{\displaystyle {\mathcal {O}}}
). Consider the minimal polynomial
f
∈
o
[
x
]
{\displaystyle f\in {\mathcal {o}}[x]}
of
θ
{\displaystyle \theta }
. The polynomial
f
¯
∈
(
o
/
p
)
[
x
]
{\displaystyle {\overline {f}}\in ({\mathcal {o}}/{\mathfrak {p}})[x]}
has the decomposition
f
¯
=
f
1
¯
e
1
⋯
f
r
¯
e
r
{\displaystyle {\overline {f}}={\overline {f_{1}}}^{e_{1}}\cdots {\overline {f_{r}}}^{e_{r}}}
with pairwise distinct irreducible polynomials
f
i
¯
{\displaystyle {\overline {f_{i}}}}
.
The factorization of
p
{\displaystyle {\mathfrak {p}}}
into prime ideals over
O
{\displaystyle {\mathcal {O}}}
is then given by
p
=
P
1
e
1
⋯
P
r
e
r
{\displaystyle {\mathfrak {p}}={\mathfrak {P}}_{1}^{e_{1}}\cdots {\mathfrak {P}}_{r}^{e_{r}}}
where
P
i
=
p
O
+
(
f
i
(
θ
)
O
)
{\displaystyle {\mathfrak {P}}_{i}={\mathfrak {p}}{\mathcal {O}}+(f_{i}(\theta ){\mathcal {O}})}
and the
f
i
{\displaystyle f_{i}}
are the polynomials
f
i
¯
{\displaystyle {\overline {f_{i}}}}
lifted to
o
[
x
]
{\displaystyle {\mathcal {o}}[x]}
.
[1]
References
^ a b Neukirch, Jürgen (1999). Algebraic number theory. Berlin: Springer. pp. 48–49. ISBN 3-540-65399-6 . OCLC 41039802.
^ Conrad, Keith. "FACTORING AFTER DEDEKIND" (PDF) .