algebraic extension of a field
1Algebraic 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.… …
2algebraic extension — Math. a field containing a given field such that every element in the first field is algebraic over the given field. Cf. extension field. * * * …
3algebraic extension — Math. a field containing a given field such that every element in the first field is algebraic over the given field. Cf. extension field …
4simple algebraic extension — Math. a simple extension in which the specified element is a root of an algebraic equation in the given field. Cf. simple transcendental extension. * * * …
5simple algebraic extension — Math. a simple extension in which the specified element is a root of an algebraic equation in the given field. Cf. simple transcendental extension …
6Field 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… …
7Field (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 …
8Algebraic number field — In mathematics, an algebraic number field (or simply number field) F is a finite (and hence algebraic) field extension of the field of rational numbers Q. Thus F is a field that contains Q and has finite dimension when considered as a vector… …
9Algebraic closure — 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.Using Zorn s lemma, it can be shown that every field has an… …
10Field 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… …
