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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 ter ...
, 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
subset In mathematics, set ''A'' is a subset of a set ''B'' if all 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 unequal, then ''A'' is a proper subset of ...
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 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 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 othe ...
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. ...
over a ground field ''K'' is ''n''. *Q( √2, ''e'') has transcendence degree 1 over Q because √2 is
algebraic Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings. Algebraic may also refer to: * Algebraic data type, a data ...
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 , \math ...
. (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 Britis ...
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 ver ...
, the field C(''S'') of meromorphic functions 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 mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
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 collection ...
. The proof that any two bases have the same cardinality depends, in each setting, on an exchange lemma. 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 matroids, 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 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
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 automorphis ...
''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 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 contrapositi ...
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 o ...
. 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''). Al ...
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 poles of the function. The ...
s on it, then trdegC(''K''(''X'')) ≤ ''n''.


See also

* Regular extension


References

Field (mathematics) Algebraic varieties Matroid theory Transcendental numbers