divisors in base field and finite extension field

divisors in base field and finite extension field

Let

k

be the

quotient field

of an

integral domain

𝔬

which has the

divisor theory

𝔬

*

→

𝔇

0

.  Let

K

/

k

a

finite extension

,

𝔒

be the

integral closure

of

𝔬

in

K

and

𝔒

*

→

𝔇

the uniquely determined divisor theory of

𝔒

(see the parent entry (

http://planetmath.org/DivisorTheoryInFiniteExtension

)).  We will study the of the

divisor

monoids

𝔇

0

and

𝔇

.

Any element

a

of

𝔬

*

, which is a part of

𝔒

*

, determines a principal divisor

(

a

)

k

∈

𝔇

0

and another

(

a

)

K

∈

𝔇

.  The (

multiplicative

) monoid

𝔬

*

is isomorphically embedded (via

ι

) in the monoid

𝔒

*

.  Because the units of the ring

𝔒

, which belong to

𝔬

, are all units of

𝔬

and because

associates

always determine the same principal divisor, the mentioned embedding defines an

isomorphic

mapping

(

a

)

k

↦

(

a

)

K

(1)

from the monoid of the principal divisors of

𝔬

into the monoid of the principal divisors of

𝔒

.  One has the

Theorem.

There is one and only one

isomorphism

φ

from the divisor monoid

𝔇

0

into the divisor monoid

𝔇

such that its restriction to the principal divisors of

𝔬

coincides with (1).  Then there is the following

commutative diagram

:

\xymatrix

⁢

𝔬

*

⁢

\ar

⁢

[

r

]

ι

⁢

\ar

⁢

[

d

]

⁢

&

⁢

𝔒

*

⁢

\ar

⁢

[

d

]

⁢

𝔇

0

⁢

\ar

⁢

[

r

]

φ

⁢

&

⁢

𝔇

The isomorphism

φ

:

𝔇

o

→

𝔇

is determined as follows.  Let

𝔭

be an arbitrary

prime divisor

in

𝔇

0

and

ν

𝔭

the corresponding

exponent valuation

of the field

k

.  Let

ν

𝔓

1

,

…

,

ν

𝔓

m

be the

continuations

of the

exponent

ν

𝔭

to

K

, which correspond to the prime divisors

𝔓

1

,

…

,

𝔓

m

in

𝔇

.  If

e

1

,

…

,

e

m

are the

ramification

indices of the exponents

ν

𝔓

1

,

…

,

ν

𝔓

m

with respect to

ν

𝔭

, then we have

ν

𝔓

i

⁢

(

a

)

=

e

i

⁢

ν

𝔭

⁢

(

a

)

∀

a

∈

𝔬

*

.

Thus apparently, the factor of the principal divisor

(

a

)

K

∈

𝔇

,  which corresponds to the factor

𝔭

ν

𝔭

⁢

(

a

)

of the principal divisor

(

a

)

k

∈

𝔇

0

, is

(

𝔓

1

e

1

⁢

⋯

⁢

𝔓

m

e

m

)

ν

𝔭

⁢

(

a

)

.  Then

φ

is settled by

𝔭

↦

𝔓

1

e

1

⁢

⋯

⁢

𝔓

m

e

m

.

When one identifies

𝔇

0

with its isomorphic image

φ

⁢

(

𝔇

0

)

, we can write

𝔭

=

𝔓

1

e

1

⁢

⋯

⁢

𝔓

m

e

m

∈

𝔇

,

i.e. the prime divisors in

𝔇

0

don’t in general remain as prime divisors in

𝔇

.  On grounds of the identification one may speak of the divisibility of the divisors of

𝔬

by the divisors of

𝔒

.  The

coprime

divisors of

𝔬

are coprime also as divisors of

𝔒

.

References

1

S. Borewicz & I. Safarevic

:

Zahlentheorie

.  Birkhäuser Verlag. Basel und Stuttgart (1966).

Title

divisors in base field and finite extension field

Canonical name

DivisorsInBaseFieldAndFiniteExtensionField

Date of creation

2013-03-22 18:01:35

Last modified on

2013-03-22 18:01:35

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

6

Author

pahio (2872)

Entry type

Topic

Classification

msc 13F05

Classification

msc 13A18

Classification

msc 13A05

Related topic

DivisorTheory

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