In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Abstract algebra, a Field extension L / K is called algebraic if every element of L is algebraic over K, i In Mathematics, a field F is said to be algebraically closed if every Polynomial in one Variable of degree at least 1 with Coefficients It is one of many closures in mathematics. In Mathematics, a set is said to be closed under some operation if the operation on members of the set produces a member of the set
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. Zorn's lemma, also known as the Kuratowski-Zorn lemma, is a proposition of Set theory that states Every Partially ordered set in which In Mathematics, the phrase " up to xxxx" indicates that members of an Equivalence class are to be regarded as a single entity for some purpose In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Mathematics, a fixed point (sometimes shortened to fixpoint) of a function is a point that is mapped to itself by the function 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. This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English.
Examples
- The fundamental theorem of algebra states that the algebraic closure of the field of real numbers is the field of complex numbers. In Mathematics, the Fundamental theorem of algebra states that every non-constant single-variable Polynomial with complex coefficients has at In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted
- The algebraic closure of the field of rational numbers is the field of algebraic numbers. In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions In Mathematics, an algebraic number is a Complex number that is a root of a non-zero Polynomial in one variable with rational (or
- 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). In Abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French (The most famous of these is the field of Nimbers, which is this field for p = 2)
- See also Puiseux expansion. In Mathematics, the Proper class of nimbers (occasionally called Grundy numbers) is introduced in Combinatorial game theory, where they are defined In Mathematics, a Puiseux expansion is a Formal power series expansion of an Algebraic function.
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. In Mathematics, an Algebraic field extension L / K is separable if it can be generated by adjoining to K a set each of whose elements In Mathematics, an Algebraic field extension L / K is separable if it can be generated by adjoining to K a set each of whose elements 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). In Mathematics, the phrase " up to xxxx" indicates that members of an Equivalence class are to be regarded as a single entity for some purpose
For K a perfect field, it is the full algebraic closure. In Mathematics, an Algebraic field extension L / K is separable if it can be generated by adjoining to K a set each of whose elements In general, the absolute Galois group of K is the Galois group of Ksep over K. In Mathematics, the absolute Galois group GK of a field K is the Galois group of K sep over K
Dapyx Software network: MP3 Explorer | Ebook Manager | Zenithic