Transcendental extension: Difference between revisions

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{{Short description|Field extension that is not algebraic}}

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''.

{{Use American English|date = January 2019}}

In [[mathematics]], a '''transcendental extension''' <math>L/K</math> is a [[field extension]] such that there exists an element in the field <math>L</math> that is [[transcendental element|transcendental]] over the field <math>K</math>; that is, an element that is not a root of any [[univariate polynomial]] with coefficients in <math>K</math>. In other words, a transcendental extension is a field extension that is not [[Algebraic extension|algebraic]]. For example, <math>\mathbb{C}</math> and <math>\mathbb{R}</math> are both transcendental extensions of <math>\mathbb{Q}.</math>

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.

A '''transcendence basis''' of a field extension <math>L/K</math> (or a transcendence basis of <math>L</math> over <math>K</math>) is a maximal [[algebraically independent]] [[subset]] of <math>L</math> over <math>K.</math> Transcendence bases share many properties with [[basis (linear algebra)|bases]] of [[vector space]]s. In particular, all transcendence bases of a field extension have the same [[cardinality]], called the '''transcendence degree''' of the extension. Thus, a field extension is a transcendental extension if and only if its transcendence degree is nonzero.

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 '''F'''_''p'' if ''L'' is of characteristic ''p''.

Transcendental extensions are widely used in [[algebraic geometry]]. For example, the [[dimension of an algebraic variety|dimension]] of an [[algebraic variety]] is the transcendence degree of its [[function field of an algebraic variety|function field]]. Also, [[global function field]]s are transcendental extensions of degree one of a [[finite field]], and play in [[number theory]] in [[positive characteristic]] a role that is very similar to the role of [[algebraic number field]]s in characteristic zero.

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'').

== Examples ==

== Transcendence basis ==

*Every algebraic extension has transcendence degree 0; the [[empty set]] serves as a transcendence basis here.

[[Zorn's lemma]] shows there exists a maximal [[linearly independent]] subset of a vector space (i.e., a basis). A similar argument with Zorn's lemma shows that, given a field extension ''L'' / ''K'', there exists a maximal algebraically independent subset of ''L'' over ''K''.<ref>{{harvnb|Milne|loc=Theorem 9.13.}}</ref> It is then called a '''transcendence basis'''. By maximality, an algebraically independent subset ''S'' of ''L'' over ''K'' is a transcendence basis if and only if ''L'' is an [[algebraic extension]] of ''K''(''S''), the field obtained by [[adjoining (field theory)|adjoining]] the elements of ''S'' to ''K''.

*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 {''x''₁,...,''x''_''n''} 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'''([[square root of two|&radic;2]], [[pi|&pi;]]) has transcendence degree 1 over '''Q''' because &radic;2 is [[algebraic number|algebraic]] while &pi; is [[transcendental number|transcendental]].

*The transcendence degree of '''C''' or '''R''' over '''Q''' is the [[continuum hypothesis|cardinality of the continuum]].

*The transcendence degree of '''Q'''(&pi;, [[E (mathematical constant)|''e'']]) is either 1 or 2; the precise answer is unknown because we don't know whether &pi; and ''e'' are algebraically independent.

The [[exchange lemma]] (a version for algebraically independent sets<ref>{{harvnb|Milne|loc=Lemma 9.6.}}</ref>) implies that if ''S'' and ''S<nowiki>'</nowiki>'' are transcendence bases, then ''S'' and ''S<nowiki>'</nowiki>'' have the same [[cardinality]]. Then the common cardinality of transcendence bases is called the '''transcendence degree''' of ''L'' over ''K'' and is denoted as <math>\operatorname{tr.deg.}_K L</math> or <math>\operatorname{tr.deg.}(L/K)</math>. There is thus an analogy: a transcendence basis and transcendence degree, on the one hand, and a basis and dimension on the other hand. 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|finitary matroids]] ([[Pregeometry (model theory)|pregeometries]]). Any finitary matroid has a basis, and all bases have the same cardinality.<ref>{{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}}.</ref>

== Analogy with vector space dimensions ==

If ''G'' is a generating set of ''L'' (i.e., ''L'' = ''K''(''G'')), then a transcendence basis for ''L'' can be taken as a subset of ''G''. Thus, <math>\operatorname{tr.deg.}_K L \le </math> the minimum cardinality of generating sets of ''L'' over ''K''. In particular, a [[finitely generated field extension]] admits a finite transcendence basis.

There is an analogy with the theory of [[vector space]] [[Hamel dimension|dimension]]s. The dictionary matches algebraically independent sets with [[linear independence|linearly independent sets]]; sets ''S'' such that ''L'' is algebraic over ''K''(''S'') with [[linear span|spanning sets]]; transcendence bases with [[Basis_(linear_algebra)|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.

If no field ''K'' is specified, the transcendence degree of a field ''L'' is its degree relative to some fixed base field; for example, the [[prime field]] of the same [[characteristic (algebra)|characteristic]], i.e.,or '''Q'K'', if ''L'' is ofan characteristic[[algebraic 0function andfield]] '''F'''_''p'' if ''L'' is of characteristicover ''pK''.

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

A '''separating transcendence basis''' of ''L'' / ''K'' is a transcendence basis ''S'' such that ''L'' is a [[separable algebraic extension]] over ''K''(''S''). A field extension ''L'' / ''K'' is said to be '''separably generated''' if it admits a separating transcendence basis.<ref>{{harvnb|Hartshorne|1977|loc=Ch I, § 4, just before Theorem 4.7.A}}</ref> If a field extension is finitely generated and it is also separably generated, then each generating set of the field extension contains a separating transcendence basis.<ref>{{harvnb|Hartshorne|1977|loc=Ch I, Theorem 4.7.A}}</ref> Over a [[perfect field]], every finitely generated field extension is separably generated; i.e., it admits a finite separating transcendence basis.<ref>{{harvnb|Milne|loc=Theorem 9.27.}}</ref>

== Examples ==

*EveryAn extension is algebraic extensionif hasand only if its transcendence degree is 0; the [[empty set]] serves as a transcendence basis here.

*The field of [[rational functionsfunction]]s in ''n'' variables ''K''(''x''₁,...,''x''_''n'') (i.e. the [[field of fractions]] of the [[polynomial ring]] ''K''[''x''₁,...,''x''_''n'']) is a purely transcendental extension with transcendence degree ''n'' over ''K''; we can for example take {''x''₁,...,''x''_''n''} as a transcendence base.

*More generally, the transcendence degree of the [[function field of an algebraic variety|function field]] ''L'' of an ''n''-dimensional [[algebraic variety]] over a ground field ''K'' is ''n''.

*'''Q'''([[square root of two|&radic;2√2]], [[piE (mathematical constant)|&pi;''e'']]) has transcendence degree 1 over '''Q''' because &radic;2√2 is [[algebraic number|algebraic]] while &pi;''e'' is [[transcendental number|transcendental]].

*The transcendence degree of '''C''' or '''R''' over '''Q''' is the [[continuum hypothesis|cardinality of the continuum]]. (Since '''Q''' is countable, the field '''Q'''(''S'') will have the same cardinality as ''S'' for any infinite set ''S'', and any algebraic extension of '''Q'''(''S'') will have the same cardinality again.)

*The transcendence degree of '''Q'''(&pi;''e'', [[Epi|π]]) (mathematicalover constant)|''e'Q''']]) is either 1 or 2; the precise answer is unknown because weit don'tis not knowknown whether &pi; and ''e'' and π are [[algebraically independent]] (see [[Schanuel's conjecture#Other consequences|Schanuel's conjecture]]).

*If ''S'' is a [[Compact space|compact]] [[Riemann surface]], the field '''C'''(''S'') of [[meromorphic function]]s on ''S'' has transcendence degree 1 over '''C'''.

== Facts ==

If ''M'' / ''L'' is a field extension and ''L'' / ''K'' is anotherare field extensionextensions, then the transcendence degree of

:trdeg(''M'' / ''K'') is equal to the sum of the transcendence degrees of= trdeg(''M'' / ''L'') and+ trdeg(''L'' / ''K''. )

This is proven by showing that a transcendence basis of ''M'' / ''K'' can be obtained by taking the [[union (set theory)|union]] of a transcendence basis of ''M'' / ''L'' and one of ''L'' / ''K''.

If the set ''S'' is algebraically independent over ''K,'' then the field ''K''(''S'') is [[isomorphic]] to the field of rational functions over ''K'' in a set of variables of the same cardinality as ''S.'' Each such rational function is a fraction of two polynomials in finitely many of those variables, with coefficients in ''K.''

Two [[algebraically closed field]]s are isomorphic if and only if they have the same characteristic and the same transcendence degree over their prime field.<ref>{{harvnb|Milne|loc=Proposition 9.16.}}</ref>

== The transcendence degree of an integral ___domain ==

Let <math>A \subseteq B</math> be [[integral ___domain]]s. If <math>Q(A)</math> and <math>Q(B)</math> denote the fields of fractions of {{math|''A''}} and {{math|''B''}}, then the ''transcendence degree'' of {{math|''B''}} over {{math|''A''}} is defined as the transcendence degree of the field extension <math>Q(B)/Q(A).</math>

The [[Noether normalization lemma]] implies that if {{math|''R''}} is an integral ___domain that is a [[finitely generated algebra]] over a field {{mvar|k}}, then the [[Krull dimension]] of {{math|''R''}} is the transcendence degree of {{math|''R''}} over {{math|''k''}}.

This has the following geometric interpretation: if {{math|''X''}} is an [[affine algebraic variety]] over a field {{math|''k''}}, the Krull dimension of its [[coordinate ring]] equals the transcendence degree of its [[function field of an algebraic variety|function field]], and this defines the [[dimension of an algebraic variety|dimension]] of {{math|''X''}}. It follows that, if {{mvar|X}} is not an affine variety, its dimension (defined as the transcendence degree of its function field) can also be defined ''locally'' as the Krull dimension of the coordinate ring of the restriction of the variety to an open affine subset.

== Relations to differentials ==

{{expand section|date=April 2023}}

Let <math>K/k</math> be a finitely generated field extension. Then<ref>{{harvnb|Hartshorne|1977|loc=Ch. II, Theorem 8.6. A}}</ref>

:<math>\dim_k \Omega_{K/k} \ge \operatorname{trdeg}(K/ k).</math>

where <math>\Omega_{K/k}</math> denotes the module of [[Kahler differential]]s. Also, in the above, the equality holds if and only if ''K'' is separably generated over ''k'' (meaning it admits a separating transcendence basis).

== Applications ==

Transcendence bases are a useful toolfor to proveproving various existence statements about [[field homomorphismshomomorphism]]s. Here is an example: Given an [[algebraically closed]] field]] ''L'', a [[Field extension|subfield]] ''K'' and a field [[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''.

As another application, we show that there are (many) proper subfields of the [[complex number|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'' &rarr;→ ''S'' which are [[injective]] but not [[surjective]]. Any such map can be extended to a field homomorphism '''Q'''(''S'') &rarr;→ '''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''' &rarr;→ '''C''' are not surjective.

The transcendence degree can give an intuitive understanding of the size of a field. For instance, a theorem due to [[Carl Ludwig Siegel|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 trdeg_'''C'''(''K''(''X''))&nbsp;≤&nbsp;''n''.

==See also==

*[[Lüroth's theorem]], a theorem about purely transcendental extensions of degree one

*[[Regular extension]]

==References==

<references/>

*{{Hartshorne AG}} <!-- Hartshorne|1977 -->

*{{citation|last=Milne|first=James|title=Field Theory|url=http://www.jmilne.org/math/CourseNotes/FT.pdf}}

*§ 6.3. of {{Citation | last1=Shimura | first1=Goro | author-link=Goro Shimura | title=Introduction to the arithmetic theory of automorphic functions | publisher=Iwanami Shoten | ___location=Tokyo | series=Publications of the Mathematical Society of Japan | year=1971 | volume=11 | zbl=0221.10029 |ref=none}}

[[Category:Field (mathematics)]]

As another application, we show that there are (many) proper subfields of the [[complex number|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'' &rarr; ''S'' which are [[injective]] but not [[surjective]]. Any such map can be extended to a field homomorphism '''Q'''(''S'') &rarr; '''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''' &rarr; '''C''' are not surjective.

[[Category:abstractAlgebraic algebravarieties]]

[[Category:Matroid theory]]

[[Category:Transcendental numbers]]