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Cale Bases in Algebraic Orders
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Let
R
be a non-maximal order in a finite algebraic number field with integral closure
R
¯
. Although
R
is not a unique factorization domain, we obtain a positive integer
N
and a family
????
(called a Cale basis) of primary irreducible elements of
R
such that
x
N
has a unique factorization into elements of
????
for each
x
∈
R
coprime with the conductor of
R
. Moreover, this property holds for each nonzero
x
∈
R
when the natural map
Spec
(
R
¯
)
→
Spec
(
R
)
is bijective. This last condition is actually equivalent to several properties linked to almost divisibility properties like inside factorial domains, almost Bézout domains, almost GCD domains.
Title: Cale Bases in Algebraic Orders
Description:
Let
R
be a non-maximal order in a finite algebraic number field with integral closure
R
¯
.
Although
R
is not a unique factorization domain, we obtain a positive integer
N
and a family
????
(called a Cale basis) of primary irreducible elements of
R
such that
x
N
has a unique factorization into elements of
????
for each
x
∈
R
coprime with the conductor of
R
.
Moreover, this property holds for each nonzero
x
∈
R
when the natural map
Spec
(
R
¯
)
→
Spec
(
R
)
is bijective.
This last condition is actually equivalent to several properties linked to almost divisibility properties like inside factorial domains, almost Bézout domains, almost GCD domains.
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