Transcendence degree
Encyclopedia
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.
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) (the field 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 and is denoted trdegK L or trdeg(L /K).
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 field extension L /K is purely transcendental if there is a subset S of L that is algebraically independent over K and such that L = K(S).
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.
of a transcendence basis of M/L and one of L /K.
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.
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: S → S 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 C → C are not surjective.
The transcendence degree can give an intuitive understanding of the size of a field. For instance, a theorem due to Siegel
states that if X is a compact, connected, complex manifold of dimension n and K(X) denotes the field of (globally defined) meromorphic function
s on it, then trdegC(K(X)) ≤ n.
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, the transcendence degree of a field extension
Field 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...
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
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
of L over K.
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
Algebraic 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 that are not algebraic, i.e...
of the field K(S) (the field 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 and is denoted trdegK L or trdeg(L /K).
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
Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...
, i.e., Q if L is of characteristic 0 and Fp if L is of characteristic p.
The field extension L /K is purely transcendental if there is a subset S of L that is algebraically independent over K and such that L = K(S).
Examples
- An extension is algebraic if and only if its transcendence degree is 0; the empty setEmpty setIn mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
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 fieldFunction field of an algebraic varietyIn algebraic geometry, the function field of an algebraic variety V consists of objects which are interpreted as rational functions on V...
L of an n-dimensional algebraic varietyAlgebraic varietyIn mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
over a ground field K is n. - Q(√2, πPi' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
) has transcendence degree 1 over Q because √2 is algebraicAlgebraic numberIn mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients. Numbers such as π that are not algebraic are said to be transcendental; almost all real numbers are transcendental...
while π is transcendentalTranscendental numberIn mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...
. - The transcendence degree of C or R over Q is the cardinality of the continuumContinuum hypothesisIn mathematics, the continuum hypothesis is a hypothesis, advanced by Georg Cantor in 1874, about the possible sizes of infinite sets. It states:Establishing the truth or falsehood of the continuum hypothesis is the first of Hilbert's 23 problems presented in the year 1900...
. (This follows since any element has only countably many algebraic elements over it in Q, since Q is itself countable.) - The transcendence degree of Q(π, eE (mathematical constant)The mathematical constant ' is the unique real number such that the value of the derivative of the function at the point is equal to 1. The function so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base...
) over Q is either 1 or 2; the precise answer is unknown because it is not known whether π and e are algebraically independent.
Analogy with vector space dimensions
There is an analogy with the theory of vector spaceVector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
dimensions. The dictionary matches algebraically independent sets with linearly independent sets
Linear independence
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. A family of vectors which is not linearly independent is called linearly dependent...
; sets S such that L is algebraic over K(S) with spanning sets
Linear span
In the mathematical subfield of linear algebra, the linear span of a set of vectors in a vector space is the intersection of all subspaces containing that set...
; transcendence bases with bases
Basis (linear algebra)
In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...
; 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.
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 unionUnion (set theory)
In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...
of a transcendence basis of M/L and one of L /K.
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 automorphismAutomorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the 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
Algebraic 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....
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.
As another application, we show that there are (many) proper subfields of the complex number field
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
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: S → S 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 C → C are not surjective.
The transcendence degree can give an intuitive understanding of the size of a field. For instance, a theorem due to Siegel
Carl Ludwig Siegel
Carl Ludwig Siegel was a mathematician specialising in number theory and celestial mechanics. He was one of the most important mathematicians of the 20th century.-Biography:...
states that if X is a compact, connected, complex manifold of dimension n and K(X) denotes the field of (globally defined) meromorphic function
Meromorphic function
In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function...
s on it, then trdegC(K(X)) ≤ n.