In abstract algebra, a field extension L /K is called algebraic if every element of L is algebraic over K, i.e. if every element of L is a root of some non-zero polynomial with coefficients in K. Field extensions which are not algebraic, i.e. which contain transcendental elements, are called transcendental. Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... In abstract algebra, an extension of a field K is a field L which contains K as a subfield. ... In mathematics, the roots of polynomials are in abstract algebra called algebraic elements. ... In mathematics, a root (or a zero) of a function f is an element x in the domain of f such that f(x) = 0. ... In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ... In mathematics, the roots of polynomials are in abstract algebra called algebraic elements. ...

For example, the field extension R/Q is transcendental, while the field extensions C/R and Q(√2)/Q are algebraic. In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit , satisfying . ...

If L is regarded as a vector space over K, one can consider its dimension as such. This dimension is also called the degree of the extension. The extension L/K can then be further classified as a finite or infinite extension according to this dimension. All transcendental extensions are of infinite degree. This in turn implies that all finite extensions are algebraic. A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... In mathematics, the dimension of a vector space V is the cardinality (i. ...

The converse is not true however: there are infinite extensions which are algebraic. For instance, the field of all algebraic numbers is an infinite algebraic extension of the rational numbers. In mathematics, an algebraic number relative to a field F is any element x of a given field K containing F such that x is a solution of a polynomial equation of the form anxn + an−1xn−1 + ··· + a1x + a0 = 0 where n is a positive integer called the degree...

If a is algebraic over K, then K[a], the set of all polynomials in a with coefficients in K, is a field. It is an algebraic field extension of K which has finite degree over K. In the special case where K=Q is the field of rational numbers, Q[a] is an example of an algebraic number field. In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics, an algebraic number field (or simply number field) is a finite field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and these days...

A field with no proper algebraic extensions is called algebraically closed. An example is the field of complex numbers. Every field has an algebraic extension which is algebraically closed (called its algebraic closure), but proving this in general requires some form of the axiom of choice. In mathematics, a field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F, has a zero in F. In that case, every such polynomial splits into linear factors. ... In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit , satisfying . ... In mathematics, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. ... In mathematics, the axiom of choice is an axiom of set theory. ...

Generalizations

Model theory generalizes the notion of algebraic extension to arbitrary theories: an embedding of M into N is called an algebraic extension if for every x in N there is a formula p with parameters in M, such that p(x) is true and the set In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the models which underlie mathematical systems. ...

{y in N | p(y)}

is finite. It turns out that applying this definition to the theory of fields gives the usual definition of algebraic extension. The Galois group of N over M can again be defined as the group of automorphisms, and it turns out that most of the theory of Galois groups can be developed for the general case. In mathematics, a Galois group is a group associated with a certain type of field extension. ...

See also:

• Abstract algebra
• Field
• Vector space
• Galois theory
• Algebraic number
• Algebraically closed field