Home > Algebraic extension

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.

For example, the field extension R/ Q is transcendental, while the field extensions C/R and Q(√2)/Q are algebraic.

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.

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.

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.

A field with no proper algebraic extensions is called

Every field has an algebraic extension which is algebraically closed (called its algebraic closureField theory In mathematics, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics. Using Zorn's lemma, it can be shown that every field has an algebraic closure, and that), but proving this in general requires some form of the axiom of choiceSet theory In mathematics, the axiom of choice is an axiom in set theory. It was formulated about a century ago by Ernst Zermelo, and was quite controversial at the time. It states the following: Stated more formally: Another formulation of the axiom of c.

Generalizations

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

{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 groupField theory Group theory In mathematics, a Galois group is a group associated with a certain type of field extension. The study of field extensions (and polynomials which give rise to them) via Galois groups is called Galois theory. Suppose E is an exten 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.

See also:

• Abstract algebra
• FieldIn abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are famil
• Vector space
• Galois theory
• Algebraic number
• Algebraically closed field