algebraically closed extension

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• 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 group — In mathematics, in the realm of group theory, a group A is algebraically closed if any finite set of equations and inequations that make sense in A already have a solution in A . This idea will be made precise later in the article.Informal… …   Wikipedia

• Quasi-algebraically 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

• 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

• 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

• Integrally closed domain — In commutative algebra, an integrally closed domain A is an integral domain whose integral closure in the field of fractions of A is A itself. Many well studied domains are integrally closed: Fields, the ring of integers Z, unique factorization… …   Wikipedia

• Field extension — In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties. For… …   Wikipedia

• Glossary of field theory — Field theory is the branch of mathematics in which fields are studied. This is a glossary of some terms of the subject. (See field theory (physics) for the unrelated field theories in physics.) Definition of a field A field is a commutative ring… …   Wikipedia

• Polynomial ring — In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the …   Wikipedia