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Encyclopedia > Transcendence degree

In abstract algebra, the transcendence degree of a field extension L / K is a certain rather coarse measure of the "size" of the extension. Specifically, it is defined as the largest cardinality of an algebraically independent subset of L over K. Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... In abstract algebra, an extension of a field K is a field L which contains K as a subfield. ... In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections, injections, and surjections, and another which uses cardinal numbers. ... In abstract algebra, a subset S of a field L is algebraically independent over a subfield K if the elements of S do not satisfy any non-trivial polynomial equation with coefficients in K. This means that for every finite sequence α1,...,αn of elements of S, no two the...


A subset S of L is a transcendence basis of L / K if it is algebraically independent over K and if furthermore L is an algebraic extension of the field K(S) obtained by adjoining the elements of S to K. One can show that every field extension has a transcendence basis, and that all transcendence bases have the same cardinality; this cardinality is equal to the transcendence degree of the extension. In abstract algebra, a field extension L /K is called algebraic if every element of L is algebraic over K, i. ...


If no field K is specified, the transcendence degree of a field L is its degree relative to the prime field of the same characteristic, i.e., Q if L is of characteristic 0 and Fp if L is of characteristic p. The word characteristic has several meanings: In mathematics, see characteristic (algebra) characteristic function characteristic subgroup Euler characteristic method of characteristics In genetics, see characteristic (genetics). ...


The field extension L / K is purely transcendental if there is a subset S of L that's algebraically independent over K and such that L = K(S).

Contents


Examples

  • Every algebraic extension has transcendence degree 0; the empty set serves as a transcendence basis here.
  • The field of rational functions in n variables K(x1,...,xn) is a purely transcendental extension with transcendence degree n over K; we can for example take {x1,...,xn} as a transcendence base
  • More generally, the transcendence degree of the function field L of an n-dimensional algebraic variety over a ground field K is n.
  • Q(√2, π) has transcendence degree 1 over Q because √2 is algebraic while π is transcendental.
  • The transcendence degree of C or R over Q is the cardinality of the continuum.
  • The transcendence degree of Q(π, e) is either 1 or 2; the precise answer is unknown because we don't know whether π and e are algebraically independent.

In mathematics, the empty set is the set with no elements. ... In algebraic geometry, the function field of an irreducible algebraic variety is the field of fractions of the ring of regular functions. ... In classical algebraic geometry (and to some extent also in modern algebraic geometry), the main objects of study are algebraic varieties. ... The square root of two is the positive real number which, when multiplied by itself, gives a product of two. ... The minuscule, or lower-case, pi The mathematical constant Ï€ represents the ratio of a circles circumference to its diameter and is commonly used in mathematics, physics, and engineering. ... In mathematics, an algebraic number relative to a field F is any element x of a given field K containing F such that x is a solution of a polynomial equation of the form anxn + an−1xn−1 + ··· + a1x + a0 = 0 where n is a positive integer called the degree... In mathematics, a transcendental number is any real number that is not algebraic, that is, not the solution of a non-zero polynomial equation with integer (or, equivalently, rational) coefficients. ... In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ... Eulers number (or Napiers constant) is the base of the natural logarithm function. ...

Analogy with vector space dimensions

There is an analogy with the theory of vector space dimensions. The dictionary matches algebraically independent sets with linearly independent sets; sets S such that L is algebraic over K(S) with spanning sets; transcendence bases with bases; and transcendence degree with dimension. The fact that transcendence bases always exist (like the fact that bases always exist in linear algebra) requires the axiom of choice. The proof that any two bases have the same cardinality depends, in each setting, on an exchange lemma. A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... In mathematics, the dimension of a vector space V is the cardinality (i. ... In linear algebra, a family of vectors is linearly independent if none of them can be written as a linear combination of finitely many other vectors in the collection. ... In the mathematical subfield of linear algebra, the linear span of a set of vectors is the set of all linear combinations of the vectors. ... In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V... In mathematics, the axiom of choice is an axiom of set theory. ...


Facts

If M/L is a field extension and L/K is another field extension, then the transcendence degree of M/K is equal to the sum of the transcendence degrees of M/L and L/K. This is proven by showing that a transcendence basis of M/K can be obtained by taking the union of a transcendence basis of M/L and one of L/K. In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ...


Applications

Transcendence bases are a useful tool to prove various existence statements about field homomorphisms. Here is an example: Given an algebraically closed field L, a subfield K and a field automorphism f of K, there exists a field automorphism of L which extends f (i.e. whose restriction to K is f). For the proof, one starts with a transcendence basis S of L/K. The elements of K(S) are just quotients of polynomials in elements of S with coefficients in K; therefore the automorphism f can be extended to one of K(S) by sending every element of S to itself. The field L is the algebraic closure of K(S) and algebraic closures are unique up to isomorphism; this means that the automorphism can be further extended from K(S) to L. In mathematics, a field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F, has a zero in F. In that case, every such polynomial splits into linear factors. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are familiar from the arithmetic of ordinary numbers. ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In mathematics, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. ...


As another application, we show that there are (many) proper subfields of the complex number field C which are (as fields) isomorphic to C. For the proof, take a transcendence basis S of C/Q. S is an infinite (even uncountable) set, so there exist (many) maps f : SS which are injective but not surjective. Any such map can be extended to a field homomorphism Q(S) → Q(S) which is not surjective. Such a field homomorphism can in turn be extended to the algebraic closure C, and the resulting field homomorphisms CC are not surjective. In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ... In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ...


  Results from FactBites:
 
Springer Online Reference Works (1083 words)
(that is, a finitely generated field extension of transcendence degree at least one; cf.
Since there is no hope for a general classification of valued function fields up to isomorphism, it makes sense to pass to their Henselizations and use the universal property of Henselizations (see Henselization of a valued field).
1) In the case of valued function fields without transcendence defect, natural criteria can be given for the isomorphism class of their Henselizations to be determined by the isomorphism classes of the value group and the residue field.
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