Algebraic closure

In 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.

Using Zorn's lemma, it can be shown that every field has an algebraic closure, and that the algebraic closure of a field "K" is unique up to an isomorphism that fixes 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 over "K" form an algebraic closure of "K".

The algebraic closure of a field "K" has the same cardinality as "K" if "K" is infinite, and is countably infinite if "K" is finite.

Examples

*The fundamental theorem of algebra states that the algebraic closure of the field of real numbers is the field of complex numbers.

*The algebraic closure of the field of rational numbers is the field of algebraic numbers.

*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 of prime order "p", the algebraic closure is a countably infinite field which contains a copy of the field of order "p""n" for each positive integer "n" (and is in fact the union of these copies). (The most famous of these is the field of Nimbers, which is this field for "p" = 2)

*See also Puiseux expansion.

eparable closure

An algebraic closure "Kalg" of "K" contains a unique separable extension "Ksep" of "K" containing all (algebraic) separable extensions 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 to isomorphism).

For "K" a perfect field, it is the full algebraic closure. In general, the absolute Galois group of "K" is the Galois group of "Ksep" over "K".


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Closure — may refer to: Closure (container) used to seal a bottle, jug, jar, can, or other container Closure (wine bottle), a stopper Closure (business), the process by which an organization ceases operations Closure (philosophy), a principle in… …   Wikipedia

  • Closure operator — In mathematics, a closure operator on a set S is a function cl: P(S) → P(S) from the power set of S to itself which satisfies the following conditions for all sets X,Y ⊆ S. X ⊆ cl(X) (cl is extensive) X ⊆ Y implies cl(X) ⊆ cl(Y)   (cl… …   Wikipedia

  • Algebraic number field — In mathematics, an algebraic number field (or simply number field) F is a finite (and hence algebraic) field extension of the field of rational numbers Q. Thus F is a field that contains Q and has finite dimension when considered as a vector… …   Wikipedia

  • 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.… …   Wikipedia

  • Algebraic number — In mathematics, an algebraic number is a complex number that is a root of a non zero polynomial in one variable with rational (or equivalently, integer) coefficients. Complex numbers such as pi that are not algebraic are said to be transcendental …   Wikipedia

  • Algebraic curve — In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with… …   Wikipedia

  • Algebraic function — In mathematics, an algebraic function is informally a function which satisfies a polynomial equation whose coefficients are themselves polynomials. For example, an algebraic function in one variable x is a solution y for an equation: a n(x)y^n+a… …   Wikipedia

  • Closure (mathematics) — For other uses, see Closure (disambiguation). In 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… …   Wikipedia

  • Closure with a twist — is a property of subsets of an algebraic structure. A subset Y of an algebraic structure X is said to exhibit closure with a twist if for every two elements there exists an automorphism ϕ of X and an element such that …   Wikipedia

  • Algebraic geometry — This Togliatti surface is an algebraic surface of degree five. Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”