function field

function field

Let

F

be a field.

Definition 1

.

The

rational function field

over

F

in one

variable

(

x

), denoted by

F

⁢

(

x

)

, is the field of all

rational functions

p

⁢

(

x

)

/

q

⁢

(

x

)

with

polynomials

p

,

q

∈

F

⁢

[

x

]

and

q

⁢

(

x

)

not identically zero.

Definition 2

.

A

function field

(in one variable) over

F

is a field

K

, containing

F

and at least one element

x

,

transcendental

over

F

, such that

K

/

F

⁢

(

x

)

is a

http://planetmath.org/node/FiniteExtension

finite

algebraic extension

.

Let

F

¯

be a fixed

algebraic closure

of

F

.

Definition 3

.

Let

K

be a function field over

F

and let

L

be a

finite extension

of

K

. The

extension

L

/

K

of function fields is said to be geometric if

L

∩

F

¯

=

F

.

Example 1

.

The extension

ℚ

⁢

(

x

)

/

ℚ

⁢

(

x

)

is geometric, but

ℚ

⁢

(

2

)

⁢

(

x

)

/

ℚ

⁢

(

x

)

is not geometric.

Theorem 1

(Thm. I.6.9 of

[

1

]

).

Let

K

be a function field over an algebraically closed field

F

. There exists a

nonsingular

projective curve

C

K

such that the function field of

C

K

is

isomorphic

to

K

.

Definition 4

.

Let

K

be a function field over a field

F

. Let

K

′

=

K

⁢

F

¯

which is a function field over

F

¯

, a fixed algebraic closure of

F

, and let

C

K

′

be the curve given by the previous theorem. The genus of

K

is, by definition, the genus of

C

K

′

.

Definition 5

.

Let

K

be a function field over a field

F

. A prime in

K

is by definition a

discrete valuation ring

R

with maximal

P

such that

F

⊂

R

and the

quotient field

of

R

is equal to

K

. The prime is usually denoted

P

after the

maximal ideal

of

R

. The degree of

P

, denoted by

deg

⁡

P

, is defined to be the

dimension

of

R

/

P

over

F

.

Example 2

.

Let

K

=

F

⁢

(

x

)

be the rational function field over

F

and let

𝒪

=

F

⁢

[

x

]

. The

prime ideals

of

𝒪

are generated by monic

irreducible polynomials

in

F

⁢

[

x

]

. Let

P

=

(

f

⁢

(

x

)

)

be such a prime. Then

R

P

=

𝒪

P

, the

localization

of

𝒪

at the prime

P

is a discrete valuation ring with

F

⊂

𝒪

P

and the quotient field of

R

P

is

K

. Thus

R

P

=

𝒪

P

is a prime of

K

.

One can define an ‘extra’ prime in the following way. Let

R

∞

=

𝒪

∞

=

F

⁢

[

1

x

]

and let

P

∞

=

(

1

x

)

be the prime ideal of

R

∞

generated by

1

x

. The localization ring

(

R

∞

)

P

∞

is a prime of

K

, called the prime at

infinity

.

References

1

R. Hartshorne,

Algebraic Geometry

, Springer-Verlag, New York.

2

M. Rosen,

Number Theory

in Function Fields

, Springer-Verlag, New York.

Title

function field

Canonical name

FunctionField

Date of creation

2013-03-22 15:34:35

Last modified on

2013-03-22 15:34:35

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

8

Author

alozano (2414)

Entry type

Definition

Classification

msc 11R58

Synonym

algebraic function field

Related topic

SimpleTranscendentalFieldExtension

Defines

rational function field

Defines

geometric extension

Defines

genus of a function field

Defines

degree of a prime

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