Algebraic element

In mathematics, if A is an associative algebra over K, then an element a of A is an algebraic element over K, or just algebraic over K, if there exists some non-zero polynomial

g

(

x

)

∈

K

[

x

]

{\displaystyle g(x)\in K[x]}

with coefficients in K such that g(a) = 0. Elements of A that are not algebraic over K are transcendental over K. A special case of an associative algebra over

K

{\displaystyle K}

is an extension field

L

{\displaystyle L}

of

K

{\displaystyle K}

. If all elements of

L

{\displaystyle L}

are algebraic over

K

{\displaystyle K}

, then

L

/

K

{\displaystyle L/K}

is called an algebraic extension.

These notions generalize the algebraic numbers and the transcendental numbers (where the field extension is C/Q, with C being the field of complex numbers and Q being the field of rational numbers).

Examples

The square root of 2 is algebraic over Q, since it is the root of the polynomial g(x) = x2 − 2 whose coefficients are rational.

Pi is transcendental over Q but algebraic over the field of real numbers R: it is the root of g(x) = x − π, whose coefficients (1 and −π) are both real, but not of any polynomial with only rational coefficients. (The definition of the term transcendental number uses C/Q, not C/R.)

An indeterminate x in a field K(x) of rational functions or K((x)) of formal Laurent series is defined to be transcendental over K.

Algebraic functions are the algebraic elements over a field of rational functions K(x1, ..., xm).

Properties

The following conditions are equivalent for an element

a

{\displaystyle a}

of an extension field

L

{\displaystyle L}

of

K

{\displaystyle K}

:

a

{\displaystyle a}

is algebraic over

K

{\displaystyle K}

,

the field extension

K

(

a

)

/

K

{\displaystyle K(a)/K}

is algebraic, i.e. every element of

K

(

a

)

{\displaystyle K(a)}

is algebraic over

K

{\displaystyle K}

(here

K

(

a

)

{\displaystyle K(a)}

denotes the smallest subfield of

L

{\displaystyle L}

containing

K

{\displaystyle K}

and

a

{\displaystyle a}

),

the field extension

K

(

a

)

/

K

{\displaystyle K(a)/K}

has finite degree, i.e. the dimension of

K

(

a

)

{\displaystyle K(a)}

as a

K

{\displaystyle K}

-vector space is finite,

K

[

a

]

=

K

(

a

)

{\displaystyle K[a]=K(a)}

, where

K

[

a

]

{\displaystyle K[a]}

is the set of all elements of

L

{\displaystyle L}

that can be written in the form

g

(

a

)

{\displaystyle g(a)}

with a polynomial

g

{\displaystyle g}

whose coefficients lie in

K

{\displaystyle K}

.