algebraically closed field

Math.a field in which every polynomial equation with coefficients that are elements of the field has at least one root in the field, as the field of complex numbers.
* * *
Universalium. 2010.
Look at other dictionaries:
Algebraically closed field — In mathematics, a field F is said to be algebraically closed if every polynomial in one variable of degree at least 1, with coefficients in F , has a root in F . ExamplesAs an example, the field of real numbers is not algebraically closed,… … Wikipedia
algebraically closed field — Math. a field in which every polynomial equation with coefficients that are elements of the field has at least one root in the field, as the field of complex numbers … Useful english dictionary
Quasialgebraically closed field — In mathematics, a field F is called quasi algebraically closed (or C1) if for every non constant homogeneous polynomial P over F has a non trivial zero provided the number of its variables is more than its degree. In other words, if P is a non… … Wikipedia
Pseudo algebraically closed field — In mathematics, a field K is pseudo algebraically closed (usually abbreviated by PAC) if one of the following equivalent conditions holds:*Each absolutely irreducible variety V defined over K has a K rational point. *Each absolutely irreducible… … Wikipedia
algebraically closed — adjective Containing all roots of single variable polynomials in its elements. According to the fundamental theorem of algebra, the field of complex numbers is algebraically closed … Wiktionary
Real closed field — In mathematics, a real closed field is a field F in which any of the following equivalent conditions are true:#There is a total order on F making it an ordered field such that, in this ordering, every positive element of F is a square in F and… … Wikipedia
Differentially closed field — In mathematics, a differential field K is differentially closed if every finite system of differential equations with a solution in some differential field extending K already has a solution in K. This concept was introduced by Robinson (1959).… … Wikipedia
Field arithmetic — In mathematics, field arithmetic is a subject that studies the interrelations between arithmetic properties of a qlfield (mathematics)field and its absolute Galois group.It is an interdisciplinary subject as it uses tools from algebraic number… … Wikipedia
Field (mathematics) — This article is about fields in algebra. For fields in geometry, see Vector field. For other uses, see Field (disambiguation). In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it … Wikipedia
Field of definition — In mathematics, the field of definition of an algebraic variety V is essentially the smallest field to which the coefficients of the polynomials defining V can belong. Given polynomials, with coefficients in a field K , it may not be obvious… … Wikipedia