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[1], 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.
Contents |
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 pn for each positive integer 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 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.
See also
Notes
- ^ M. F. Atiyah and I. G. Macdonald (1969). Introduction to commutative algebra. Addison-Wesley publishing Company. pp. 11-12.
Open source encyclopedia content modification information:
This page was last modified on 17 March 2010 at 02:17.
Authorship and Review
Open source encyclopedia content provided here is not reviewed directly by MedLibrary.org. Content is sourced directly from Wikipedia and is authored by an open community of volunteers. It is not produced by or in any way affiliated with MedLibrary.org.
Usage Guidelines
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article on "Algebraic closure", which is available in its original form here:
http://en.wikipedia.org/w/index.php?title=Algebraic_closure
All material adapted used from Wikipedia is available under the terms of the GNU Free Documentation License. Wikipedia® itself is a registered trademark of the Wikimedia Foundation, Inc.