Algebraic element

# Algebraic element

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In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, if L is a field extension
Field extension
In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties...

of K, then an element a of L is called an algebraic element over K, or just algebraic over K, if there exists some non-zero polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

g(x) with coefficient
Coefficient
In mathematics, a coefficient is a multiplicative factor in some term of an expression ; it is usually a number, but in any case does not involve any variables of the expression...

s in K such that g(a)=0. Elements of L which are not algebraic over K are called transcendental over K.

These notions generalize the algebraic number
Algebraic number
In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients. Numbers such as π that are not algebraic are said to be transcendental; almost all real numbers are transcendental...

s and the transcendental number
Transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...

s (where the field extension is C/Q, C being the field of complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

s and Q being the field of rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

s).

## Examples

• The square root
Square root
In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...

of 2 is algebraic over Q, since it is the root of the polynomial g(x) = x2 - 2 whose coefficients are rational.
• Pi
Pi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...

is transcendental over Q but algebraic over the field of real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s 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
Transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...

uses C/Q, not C/R.)

## Properties

The following conditions are equivalent for an element a of L:
• a is algebraic over K
• the field extension K(a)/K has finite degree, i.e. the dimension of K(a) as a K-vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

is finite. (Here K(a) denotes the smallest subfield of L containing K and a)
• K[a] = K(a), where K[a] is the set of all elements of L that can be written in the form g(a) with a polynomial g whose coefficients lie in K.

This characterization can be used to show that the sum, difference, product and quotient of algebraic elements over K are again algebraic over K. The set of all elements of L which are algebraic over K is a field that sits in between L and K.

If a is algebraic over K, then there are many non-zero polynomials g(x) with coefficients in K such that g(a) = 0. However there is a single one with smallest degree and with leading coefficient 1. This is the minimal polynomial of a and it encodes many important properties of a.

Fields that do not allow any algebraic elements over them (except their own elements) are called algebraically closed
Algebraically closed field
In mathematics, a field F is said to be algebraically closed if every polynomial with one variable of degree at least 1, with coefficients in F, has a root in F.-Examples:...

. The field of complex numbers is an example.