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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.
In mathematics, particularly in algebra, a field extension is a pair of fields such that the operations of K are those of L restricted to K. In this case, L is an extension field of K and K is a subfield of L. For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers.
In mathematics, a transcendental extension is a field extension such that there exists an element in the field that is transcendental over the field ; that is, an element that is not a root of any univariate polynomial with coefficients in . In other words, a transcendental extension is a field extension that is not algebraic. For example, are both transcendental extensions of
Field theory is the branch of mathematics in which fields are studied. This is a glossary of some terms of the subject.
In mathematics, a polynomial P(X) over a given field K is separable if its roots are distinct in an algebraic closure of K, that is, the number of distinct roots is equal to the degree of the polynomial.
In field theory, a branch of algebra, an algebraic field extension is called a separable extension if for every , the minimal polynomial of over F is a separable polynomial. There is also a more general definition that applies when E is not necessarily algebraic over F. An extension that is not separable is said to be inseparable.
In mathematics, an algebraic torus, where a one dimensional torus is typically denoted by , , or , is a type of commutative affine algebraic group commonly found in projective algebraic geometry and toric geometry. Higher dimensional algebraic tori can be modelled as a product of algebraic groups . These groups were named by analogy with the theory of tori in Lie group theory. For example, over the complex numbers the algebraic torus is isomorphic to the group scheme , which is the scheme theoretic analogue of the Lie group . In fact, any -action on a complex vector space can be pulled back to a -action from the inclusion as real manifolds.
In algebra, a field k is perfect if any one of the following equivalent conditions holds:
In mathematics, a field F is called quasi-algebraically closed if every non-constant homogeneous polynomial P over F has a non-trivial zero provided the number of its variables is more than its degree. The idea of quasi-algebraically closed fields was investigated by C. C. Tsen, a student of Emmy Noether, in a 1936 paper ; and later by Serge Lang in his 1951 Princeton University dissertation and in his 1952 paper. The idea itself is attributed to Lang's advisor Emil Artin.
In mathematics, the absolute Galois groupGK of a field K is the Galois group of Ksep over K, where Ksep is a separable closure of K. Alternatively it is the group of all automorphisms of the algebraic closure of K that fix K. The absolute Galois group is well-defined up to inner automorphism. It is a profinite group.
In mathematics, a thin set in the sense of Serre, named after Jean-Pierre Serre, is a certain kind of subset constructed in algebraic geometry over a given field K, by allowed operations that are in a definite sense 'unlikely'. The two fundamental ones are: solving a polynomial equation that may or may not be the case; solving within K a polynomial that does not always factorise. One is also allowed to take finite unions.
In mathematics, a field is pseudo algebraically closed if it satisfies certain properties which hold for algebraically closed fields. The concept was introduced by James Ax in 1967.
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 whether there is a smaller field k, and other polynomials defined over k, which still define V.
In commutative algebra, an element b of a commutative ring B is said to be integral overA, a subring of B, if there are n ≥ 1 and aj in A such that
In mathematics, a separable algebra is a kind of semisimple algebra. It is a generalization to associative algebras of the notion of a separable field extension.
In mathematics, a Henselian ring is a local ring in which Hensel's lemma holds. They were introduced by Azumaya (1951), who named them after Kurt Hensel. Azumaya originally allowed Henselian rings to be non-commutative, but most authors now restrict them to be commutative.
In mathematics, field arithmetic is a subject that studies the interrelations between arithmetic properties of a field and its absolute Galois group. It is an interdisciplinary subject as it uses tools from algebraic number theory, arithmetic geometry, algebraic geometry, model theory, the theory of finite groups and of profinite groups.
In mathematics, the Haran diamond theorem gives a general sufficient condition for a separable extension of a Hilbertian field to be Hilbertian.
In field theory, a branch of algebra, a field extension is said to be regular if k is algebraically closed in L and L is separable over k, or equivalently, is an integral domain when is the algebraic closure of .
Moshe Jarden is an Israeli mathematician, specialist in field arithmetic.
References
- Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Vol. 11 (3rd revised ed.). Springer-Verlag. pp. 38–44. ISBN 978-3-540-77269-9. Zbl 1145.12001.
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