Algebraic independence

In abstract algebra, a subset of a field is algebraically independent over a subfield if the elements of do not satisfy any non-trivial polynomial equation with coefficients in . In particular, a one element set is algebraically independent over if and only if is transcendental over . In general, all the elements of an algebraically independent set over are by necessity transcendental over , and over all of the field extensions over generated by the remaining elements of . The two real numbers and are each transcendental numbers: they are not the roots of any nontrivial polynomial whose coefficients are rational numbers. Thus, each of the two singleton sets and are algebraically independent over the field of rational numbers. However, the set is not algebraically independent over the rational numbers, because the nontrivial polynomial is zero when and . Although both and e are known to be transcendental, it is not known whether the set of both of them is algebraically independent over . In fact, it is not even known if is irrational. Nesterenko proved in 1996 that: the numbers , , and , where is the gamma function, are algebraically independent over . the numbers and are algebraically independent over . for all positive integers , the number is algebraically independent over . The Lindemann–Weierstrass theorem can often be used to prove that some sets are algebraically independent over . It states that whenever are algebraic numbers that are linearly independent over , then are also algebraically independent over . Algebraic matroid Given a field extension which is not algebraic, Zorn's lemma can be used to show that there always exists a maximal algebraically independent subset of over . Further, all the maximal algebraically independent subsets have the same cardinality, known as the transcendence degree of the extension. For every set of elements of , the algebraically independent subsets of satisfy the axioms that define the independent sets of a matroid.

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Matroid rank

In the mathematical theory of matroids, the rank of a matroid is the maximum size of an independent set in the matroid. The rank of a subset S of elements of the matroid is, similarly, the maximum size of an independent subset of S, and the rank function of the matroid maps sets of elements to their ranks. The rank function is one of the fundamental concepts of matroid theory via which matroids may be axiomatized. Matroid rank functions form an important subclass of the submodular set functions.

Real number

In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature. Here, continuous means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives.

Transcendental extension

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 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.

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