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. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... In 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 same rules hold which are familiar from the arithmetic of ordinary numbers. ... In abstract algebra, a field extension L /K is called algebraic if every element of L is algebraic over K, i. ... The word that is used in the English language for several grammatical purposes: to introduce a restrictive clause as a demonstrative pronoun This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... In mathematics, a field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F, has a zero (root) in F (i. ... In mathematics, the closure C(X) of an object X is defined to be the smallest object that both includes X as a subset and possesses some given property. ...

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. Zorns lemma, also known as the Kuratowski-Zorn lemma, is a theorem of set theory that states: Every partially ordered set in which every chain (i. ... Look up Up to in Wiktionary, the free dictionary Modern Slang In modern slang, up to means you are either willing to engage in an act (Sally is up to going to the park), capable of an act (Im sorry, Im just not up to it) or are... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... In mathematics, a fixed point of a function is a point that is mapped to itself by the function. ...

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. In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal numbers, words that are used for order (first, second, third). ... In mathematics the term countable set is used to describe the size of a set, e. ...

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).