In
abstract algebraAbstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, a
field extensionIn 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...
L/
K is called
algebraic if every element of
L is
algebraicIn mathematics, if L is a field extension 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 g with coefficients in K such that g=0...
over
K, i.e. if every element of
L is a root of some non-zero
polynomialIn 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...
with coefficients in
K. Field extensions that are not algebraic, i.e. which contain transcendental elements, are called
transcendental.
For example, the field extension
R/
Q, that is the field of
real numberIn 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 as an extension of the field of
rational numberIn 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, is transcendental, while the field extensions
C/
R and
Q(√2)/
Q are algebraic, where
C is the field of
complex numberA 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.
All transcendental extensions are of
infinite degreeIn mathematics, more specifically field theory, the degree of a field extension is a rough measure of the "size" of the extension. The concept plays an important role in many parts of mathematics, including algebra and number theory — indeed in any area where fields appear prominently.-...
. 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 numberIn 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 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 not only a ring but a field: an algebraic extension of
K which has finite degree over
K. In the special case where
K =
Q is the
field of rational numbersIn 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...
,
Q[
a] is an example of an
algebraic number fieldIn mathematics, an algebraic number field F is a finite field extension of the field of rational numbers Q...
.
A field with no proper algebraic extensions is called
algebraically closedIn 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:...
. An example is the field of
complex numberA 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. Every field has an algebraic extension which is algebraically closed (called its
algebraic closureIn mathematics, particularly abstract algebra, 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....
), but proving this in general requires some form of the
axiom of choice.
An extension
L/
K is algebraic
if and only ifIn logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
every sub
K-algebra of
L is a
fieldIn abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
.
Generalizations
Model theoryIn mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
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
- {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 groupIn mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...
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.