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, and are both transcendental extensions of
A transcendence basis of a field extension (or a transcendence basis of over ) is a maximal algebraically independent subset of over Transcendence bases share many properties with bases of vector spaces. In particular, all transcendence bases of a field extension have the same cardinality, called the transcendence degree of the extension. Thus, a field extension is a transcendental extension if and only if its transcendence degree is nonzero.
Transcendental extensions are widely used in algebraic geometry. For example, the dimension of an algebraic variety is the transcendence degree of its function field. Also, global function fields are transcendental extensions of degree one of a finite field, and play in number theory in positive characteristic a role that is very similar to the role of algebraic number fields in characteristic zero.
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In mathematics, a transcendentalextension L / K {\displaystyle L/K} is a field extension such that there exists an element in the field L {\displaystyle...
purely transcendental—a class of such extensions take the form L/K where both L and K are algebraically closed. If L/K is purely transcendental and S is...
a transcendentalextension, it is natural to use the fact that every field extension is an algebraic extension of a purely transcendentalextension. This...
coefficients in K. A field extension that is not algebraic, is said to be transcendental, and must contain transcendental elements, that is, elements...
transcendental numbers are π and e. The quality of a number being transcendental is called transcendence. Though only a few classes of transcendental...
field extension C(T)/C, where C(T) is the field of rational functions over C, has infinite degree (indeed it is a purely transcendentalextension). This...
The Transcendental Meditation technique (abbreviated as TM) is the technique associated with the practice of Transcendental Meditation developed by the...
isomorphism that fixes F. Transcendental Those elements of an extension field of F that are not algebraic over F are transcendental over F. Algebraically...
transcendental element, an element of a field extension that is not the root of any polynomial with coefficients from the base field Transcendental function...
or general system of ends. The speculative extension of reason is severely limited in the transcendental dialectics of the Critique of Pure Reason, which...
are called transcendental over K. These notions generalize the algebraic numbers and the transcendental numbers (where the field extension is C/Q, with...
extensions are also described as purely transcendental. More precisely, the rationality question for the field extension K ⊂ L {\displaystyle K\subset L} is...
Transcendental number theory is a branch of number theory that investigates transcendental numbers (numbers that are not solutions of any polynomial equation...
K[y1,..., ym]. Thus the extension K ⊂ A can be written as a composite K ⊂ B ⊂ A where K ⊂ B is a purely transcendentalextension and B ⊂ A is finite. In...
permutation group Sn acting on the field k(x1, ..., xn) always is a pure transcendentalextension of the field k. (She first mentioned this problem in a 1913 paper...
is transcendental over K, K ( X ) ( X p ) ⊃ K ( X ) {\displaystyle K(X)({\sqrt[{p}]{X}})\supset K(X)} is a non-separable algebraic field extension. In...
indeterminates over k {\displaystyle k} , or in other words a pure transcendentalextension of k {\displaystyle k} , with transcendence degree n {\displaystyle...
solutions used in extensions of the algebra. By starting with the field of rational functions, two special types of transcendentalextensions (the logarithm...
closed field is regular. An extension is regular if and only if it is separable and primary. A purely transcendentalextension of a field is regular. There...
two different kinds of simple extensions (see Structure of simple extensions below). The element θ may be transcendental over K, which means that it is...
The Transcendental Meditation movement (TM) are programs and organizations that promote the Transcendental Meditation technique founded by Maharishi Mahesh...
Advanced Vector Extensions (AVX, also known as Gesher New Instructions and then Sandy Bridge New Instructions) are SIMD extensions to the x86 instruction...
{\displaystyle K} are by necessity transcendental over K {\displaystyle K} , and over all of the field extensions over K {\displaystyle K} generated by...
his retirement in 1987. Deleuze's outlook on life was sympathetic to transcendental ideas, "nature as god" ethics, and the monist experience. Some of the...
in a purely transcendentalextension to the multivariate case over the ground field (see below). From coefficients in an algebraic extension to coefficients...
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