examples of fields

examples of fields

Fields (

http://planetmath.org/Field

) are typically sets of “numbers” in which the

arithmetic
operations

of

addition

,

subtraction

,

multiplication

and

division

are
defined. The following is a list of

examples of fields

.

•

The

set of all rational numbers

ℚ

, all

real numbers

ℝ

and all

complex numbers

ℂ

are the most familiar examples of fields.

•

Slightly more exotic, the

hyperreal numbers

and the

surreal
numbers

are fields containing

infinitesimal

and infinitely large
numbers. (The surreal numbers aren’t a field in the strict sense since
they form a

proper class

and not a set.)

•

The

algebraic numbers

form a field; this is the

algebraic
closure

of

ℚ

. In general, every field has an (essentially
unique) algebraic closure.

•

The

computable

complex numbers (those whose digit

sequence

can be produced by a

Turing machine

) form a field. The

definable

complex numbers (those which can be
precisely specified using a logical

formula

) form a field containing the computable numbers; arguably, this
field contains all the numbers we can ever talk about. It is

countable

.

•

The so-called

algebraic number fields

(sometimes just called

number fields

) arise from

ℚ

by adjoining some (finite number of) algebraic numbers. For

instance

ℚ

⁢

(

2

)

=

{

u

+

v

⁢

2

∣

u

,

v

∈

ℚ

}

and

ℚ

⁢

(

2

3

,

i

)

=

{

u

+

v

⁢

i

+

w

⁢

2

3

+

x

⁢

i

⁢

2

3

+

y

⁢

4

3

+

z

⁢

i

⁢

4

3

∣

u

,

v

,

w

,

x

,

y

,

z

∈

ℚ

}

=

ℚ

⁢

(

i

⁢

2

3

)

(every

separable

finite field extension

is simple).

•

If

p

is a

prime number

, then the

p

-adic numbers

form a
field

ℚ

p

which is the

completion

of the field

ℚ

with respect to the

p

-adic

valuation

.

•

If

p

is a prime number, then the

integers

modulo

p

form a

finite field

with

p

elements

, typically denoted by

𝔽

p

. More
generally, for every prime (

http://planetmath.org/Prime

) power

p

n

there is one and only one
finite field

𝔽

p

n

with

p

n

elements.

•

If

K

is a field, we can form the

field of rational functions

over

K

, denoted by

K

⁢

(

X

)

. It consists of quotients of

polynomials

in

X

with coefficients in

K

.

•

If

V

is a

variety

(

http://planetmath.org/AffineVariety

) over the field

K

, then the

function field

of

V

, denoted by

K

⁢

(

V

)

, consists of all quotients of

polynomial functions

defined on

V

.

•

If

U

is a domain (= connected open set) in

ℂ

, then the
set of all

meromorphic functions

on

U

is a field. More generally, the meromorphic functions on any

Riemann surface

form a field.

•

If

X

is a variety (or scheme) then the

rational functions

on

X

form a field. At each point of

X

, there is also a

residue field

which contains information about that point.

•

The field of

formal Laurent series

over the field

K

in the
variable

X

consists
of all expressions of the form

∑

j

=

-

M

∞

a

j

⁢

X

j

where

M

is some integer and the coefficients

a

j

come from

K

.

•

More generally, whenever

R

is an

integral domain

, we can form
its

field of fractions

, a field whose elements are the

fractions

of

elements of

R

.

Many of the fields described above have some sort of additional

structure

, for example a topology (yielding a

topological field

), a total order, or a canonical

absolute value

.

Title

examples of fields

Canonical name

ExamplesOfFields

Date of creation

2013-03-22 12:50:13

Last modified on

2013-03-22 12:50:13

Owner

AxelBoldt (56)

Last modified by

AxelBoldt (56)

Numerical id

16

Author

AxelBoldt (56)

Entry type

Example

Classification

msc 12E99

Related topic

NumberField

Generated on Fri Feb 9 16:41:20 2018 by

LaTeXML