In
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
, the transcendence degree of a
field extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
''L'' / ''K'' is a certain rather coarse measure of the "size" of the extension. Specifically, it is defined as the largest
cardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
of an
algebraically independent
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.
In particular, a one element set \ is algebraically in ...
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
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
In mathematics, an algebraic extension is a field extension such that every element of the larger field is algebraic over the smaller field ; that is, if every element of is a root of a non-zero polynomial with coefficients in . A field ext ...
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 trdeg
''K'' ''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
In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive ide ...
of the same
characteristic, i.e., the rational numbers field Q if ''L'' is of characteristic 0 and the finite field F
''p'' 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 set
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
serves as a transcendence basis here.
*The field of rational functions in ''n'' variables ''K''(''x''
1,...,''x''
''n'') is a purely transcendental extension with transcendence degree ''n'' over ''K''; we can for example take as a transcendence base.
*More generally, the transcendence degree of the
function field ''L'' of an ''n''-dimensional
algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...
over a ground field ''K'' is ''n''.
*Q(
√2,
''e'') has transcendence degree 1 over Q because √2 is
algebraic while ''e'' is
transcendental.
*The transcendence degree of C or R over Q is the
cardinality of the continuum
In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers \mathbb R, sometimes called the continuum. It is an infinite cardinal number and is denoted by \mathfrak c (lowercase fraktur "c") or , \mathb ...
. (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(''e'',
π) over Q is either 1 or 2; the precise answer is unknown because it is not known whether ''e'' and π are algebraically independent.
*If ''S'' is a
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
, the field C(''S'') of
meromorphic functions
In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are pole (complex analysis), pole ...
on ''S'' has transcendence degree 1 over C.
Analogy with vector space dimensions
There is an analogy with the theory of
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
s. The analogy 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
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collectio ...
. The proof that any two bases have the same cardinality depends, in each setting, on an
exchange lemma
The Steinitz exchange lemma is a basic theorem in linear algebra used, for example, to show that any two bases for a finite- dimensional vector space have the same number of elements. The result is named after the German mathematician Ernst Steini ...
.
This analogy can be made more formal, by observing that linear independence in vector spaces and algebraic independence in field extensions both form examples of
matroid
In combinatorics, a branch of mathematics, a matroid is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid axiomatically, the most significant being ...
s, called linear matroids and algebraic matroids respectively. Thus, the transcendence degree is the
rank function of an algebraic matroid. Every linear matroid is isomorphic to an algebraic matroid, but not vice versa.
[{{citation, title=Applied Discrete Structures, first=K. D., last=Joshi, publisher=New Age International, year=1997, isbn=9788122408263, page=909, url=https://books.google.com/books?id=lxIgGGJXacoC&pg=PA909.]
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
Union commonly refers to:
* Trade union, an organization of workers
* Union (set theory), in mathematics, a fundamental operation on sets
Union may also refer to:
Arts and entertainment
Music
* Union (band), an American rock group
** ''Un ...
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
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in .
Examples
As an example, the field of real numbers is not algebraically closed, because ...
field ''L'', a
subfield ''K'' and a field
automorphism
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 ...
''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
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.
Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky ( ...
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
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
but not
surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
. 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
Siegel (also Segal or Segel), is a German and Ashkenazi Jewish surname. it can be traced to 11th century Bavaria and was used by people who made wax seals for or sealed official documents (each such male being described as a ''Siegelbeamter''). ...
states that if ''X'' is a compact, connected, complex manifold of dimension ''n'' and ''K''(''X'') denotes the field of (globally defined)
meromorphic function
In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are pole (complex analysis), pole ...
s on it, then trdeg
C(''K''(''X'')) ≤ ''n''.
See also
*
Regular extension
References
Field (mathematics)
Algebraic varieties
Matroid theory
Transcendental numbers