In
mathematicsMathematics 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...
, particularly
abstract algebraAbstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, an
algebraic closure of 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...
K is an
algebraic extensionIn 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 that are not algebraic, i.e...
of
K that is
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:...
. It is one of many
closuresIn mathematics, a set is said to be closed under some operation if performance of that operation on members of the set always produces a unique member of the same set. For example, the real numbers are closed under subtraction, but the natural numbers are not: 3 and 8 are both natural numbers, but...
in mathematics.
Using
Zorn's lemmaZorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory that states:Suppose a partially ordered set P has the property that every chain has an upper bound in P...
, it can be shown that every field has an algebraic closure, and that the algebraic closure of a field
K is unique
up toIn mathematics, the phrase "up to x" means "disregarding a possible difference in x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...
an
isomorphismIn abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
that
fixesIn mathematics, a fixed point of a function is a point that is mapped to itself by the function. A set of fixed points is sometimes called a fixed set...
every member of
K. Because of this essential uniqueness, we often speak of
the algebraic closure of
K, rather than
an algebraic closure of
K.
The algebraic closure of a field
K can be thought of as the largest algebraic extension of
K.
To see this, note that if
L is any algebraic extension of
K, then the algebraic closure of
L is also an algebraic closure of
K, and so
L is contained within the algebraic closure of
K.
The algebraic closure of
K is also the smallest algebraically closed field containing
K,
because if
M is any algebraically closed field containing
K, then the elements of
M which are
algebraic overIn 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 that are not algebraic, i.e...
K form an algebraic closure of
K.
The algebraic closure of a field
K has the same
cardinalityIn mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...
as
K if
K is infinite, and is countably infinite if
K is finite.
Examples
- The fundamental theorem of algebra
The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root...
states that the algebraic closure of 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 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.
- The algebraic closure of the field of 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 is the field of 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.
- There are many countable algebraically closed fields within the complex numbers, and strictly containing the field of algebraic numbers; these are the algebraic closures of transcendental extensions of the rational numbers, e.g. the algebraic closure of Q(π).
- For a finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
of primeA prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
order p, the algebraic closure is a countably infinite field which contains a copy of the field of order pn for each positive integerThe integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
n (and is in fact the union of these copies).
- See also Puiseux expansion.
Separable closure
An algebraic closure
Kalg of
K contains a unique
separable extensionIn modern algebra, an algebraic field extension E\supseteq F is a separable extension if and only if for every \alpha\in E, the minimal polynomial of \alpha over F is a separable polynomial . Otherwise, the extension is called inseparable...
Ksep of
K containing all (algebraic)
separable extensionIn modern algebra, an algebraic field extension E\supseteq F is a separable extension if and only if for every \alpha\in E, the minimal polynomial of \alpha over F is a separable polynomial . Otherwise, the extension is called inseparable...
s of
K within
Kalg. This subextension is called a
separable closure of
K. Since a separable extension of a separable extension is again separable, there are no finite separable extensions of
Ksep, of degree > 1. Saying this another way,
K is contained in a
separably-closed algebraic extension field. It is essentially unique (
up toIn mathematics, the phrase "up to x" means "disregarding a possible difference in x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...
isomorphism).
The separable closure is the full algebraic closure if and only if
K is a
perfect fieldIn algebra, a field k is said to be perfect if any one of the following equivalent conditions holds:* Every irreducible polynomial over k has distinct roots.* Every polynomial over k is separable.* Every finite extension of k is separable...
. For example, if
K is a field of characteristic
p and if
X is transcendental over
K,
is a non-separable algebraic field extension.
In general, the
absolute Galois groupIn mathematics, the absolute Galois group GK of a field K is the Galois group of Ksep over K, where Ksep is a separable closure of K. Alternatively it is the group of all automorphisms of the algebraic closure of K that fix K. The absolute Galois group is unique up to isomorphism...
of
K is the Galois group of
Ksep over
K.