separable field
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Separable polynomial — In mathematics, two slightly different notions of separable polynomial are used, by different authors. According to the most common one, a polynomial P(X) over a given field K is separable if all its roots are distinct in an algebraic closure of… … Wikipedia
Separable algebra — A separable algebra is a kind of semisimple algebra. It is a generalization to associative algebras of the notion of a separable field extension. Definition Let K be a field. An associative K algebra A is said to be separable if for every field… … 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 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
Separable extension — 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 is a root of a separable polynomial over K . In that case, each beta; in L has a separable minimal polynomial… … 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
Separable sigma algebra — In mathematics, sigma; algebras are usually studied in the context of measure theory. A separable sigma; algebra (or separable sigma; field) is a sigma algebra that can be generated by a countable collection of sets. To learn what is meant by the … Wikipedia
Perfect field — In algebra, a field k is said to be perfect if any one of the following equivalent conditions holds: Every irreducible polynomial over k has distinct roots. Every polynomial over k is separable. Every finite extension of k is separable. (This… … Wikipedia
Finite field — 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. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and… … Wikipedia
Splitting field — In abstract algebra, a splitting field of a polynomial with coefficients in a field is a smallest field extension of that field over which the polynomial factors (or splits , hence the name) into linear factors. Contents 1 Definition 2 Facts 3 … 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
