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A field extension that is not algebraic (cf. [[Extension of a field|Extension of a field]]). An extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t0936201.png" /> is transcendental if and only if the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t0936202.png" /> contains elements that are transcendental over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t0936203.png" />, that is, elements that are not roots of any non-zero polynomial with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t0936204.png" />.
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{{TEX|done}}
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{{MSC|12Fxx}}
  
The elements of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t0936205.png" /> are called algebraically independent over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t0936206.png" /> if for any finite set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t0936207.png" /> and any non-zero polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t0936208.png" /> with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t0936209.png" />,
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A transcendental extension of a field $k$ is
 +
a field extension that is not algebraic (cf.
 +
[[Extension of a field|Extension of a field]]). An extension $K/k$ is
 +
transcendental if and only if the field $K$ contains elements that are
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transcendental over $k$, that is, elements that are not roots of any
 +
non-zero polynomial with coefficients in $k$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362010.png" /></td> </tr></table>
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The elements of a set $X\subset K$ are called algebraically independent over
 +
$k$ if for any finite set $x_1,\dots,x_m \in X$ and any non-zero polynomial $F(X_1,\dots,X_m)$ with
 +
coefficients in $k$,
 +
$$F(x_1,\dots,x_m)\ne 0.$$
 +
The elements of $X$ are transcendental over
 +
$k$. If $X\subset K$ is a maximal set of algebraically independent elements
 +
over $k$, then $X$ is called a transcendence basis of $K$ over
 +
$k$. The cardinality of $X$ is called the transcendence degree of $K$
 +
over $k$ and is an invariant of the extension $K/k$. For a
 +
[[Tower of fields|tower of fields]] $L\supset K\supset k$, the transcendence degree of
 +
$L/k$ is equal to the sum of the transcendence degrees of $L/K$ and $K/k$.
  
The elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362011.png" /> are transcendental over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362012.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362013.png" /> is a maximal set of algebraically independent elements over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362014.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362015.png" /> is called a transcendence basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362016.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362017.png" />. The cardinality of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362018.png" /> is called the transcendence degree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362019.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362020.png" /> and is an invariant of the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362021.png" />. For a [[Tower of fields|tower of fields]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362022.png" />, the transcendence degree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362023.png" /> is equal to the sum of the transcendence degrees of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362025.png" />.
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If all elements of a set $X$ are algebraically independent over $k$,
 +
then the extension $k(X)$ is called purely transcendental. In this case
 +
the field $k(X)$ is isomorphic to the field of rational functions in the
 +
set of variables $X$ over $k$. Any field extension $L/k$ can be
 +
represented as a tower of extensions $L\supset K\supset k$, where $L/K$ is an algebraic
 +
and $K/k$ is a purely transcendental extension. If $K$ can be chosen so
 +
that $L/K$ is a
 +
[[Separable extension|separable extension]], then the extension $K/k$ is
 +
called separably generated, and the transcendence basis of $K$ over
 +
$k$ is called a separating basis. If $L$ is separably generated over
 +
$k$, then $L$ is separable over $k$. In the case when the extension
 +
$L/k$ is finitely generated, the converse holds as well. By definition,
 +
an extension $K/k$ is separable if and only if any derivation (cf.
 +
[[Derivation in a ring|Derivation in a ring]]) of $k$ extends to
 +
$K$. Such an extension is uniquely determined for any derivation if
 +
and only if the extension $K/k$ is algebraic.
  
If all elements of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362026.png" /> are algebraically independent over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362027.png" />, then the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362028.png" /> is called purely transcendental. In this case the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362029.png" /> is isomorphic to the field of rational functions in the set of variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362030.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362031.png" />. Any field extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362032.png" /> can be represented as a tower of extensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362033.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362034.png" /> is an algebraic and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362035.png" /> is a purely transcendental extension. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362036.png" /> can be chosen so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362037.png" /> is a [[Separable extension|separable extension]], then the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362038.png" /> is called separably generated, and the transcendence basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362039.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362040.png" /> is called a separating basis. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362041.png" /> is separably generated over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362042.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362043.png" /> is separable over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362044.png" />. In the case when the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362045.png" /> is finitely generated, the converse holds as well. By definition, an extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362046.png" /> is separable if and only if any derivation (cf. [[Derivation in a ring|Derivation in a ring]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362047.png" /> extends to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362048.png" />. Such an extension is uniquely determined for any derivation if and only if the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362049.png" /> is algebraic.
+
====References====
 +
{|
 +
|-
 +
|valign="top"|{{Ref|Bo}}||valign="top"| N. Bourbaki, "Algebra", ''Elements of mathematics'', '''1''', Springer (1988) pp. Chapt. 4–6 (Translated from French)
 +
|-
 +
|valign="top"|{{Ref|ZaSa}}||valign="top"| O. Zariski, P. Samuel, "Commutative algebra", '''1''', Springer (1975)
 +
|-
 +
|}
  
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  O. Zariski,  P. Samuel,  "Commutative algebra" , '''1''' , Springer  (1975)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Bourbaki,  "Algebra" , ''Elements of mathematics'' , '''1''' , Springer  (1988)  pp. Chapt. 4–6  (Translated from French)</TD></TR></table>
 
  
  
  
 
====Comments====
 
====Comments====
The Noether normalization lemma says that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362050.png" /> is an integral domain that is finitely generated as a ring over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362051.png" />, then there exist <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362052.png" /> that are algebraically independent over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362053.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362054.png" /> is integral over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093620/t09362055.png" />.
+
The Noether normalization lemma says that if $A$ is
 +
an integral domain that is finitely generated as a ring over a field
 +
$k$, then there exist $x_1,\dots,x_r \in A$ that are algebraically independent over $k$
 +
such that $A$ is integral over $k[x_1,\dots,x_r]$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"P.M. Cohn,   "Algebra" , '''1–2''' , Wiley (1989) pp. Vol. 2, 350; Vol. 3, 168ff</TD></TR></table>
+
{|
 +
|-
 +
|valign="top"|{{Ref|Co}}||valign="top"| P.M. Cohn, "Algebra", '''1–2''', Wiley (1989) pp. Vol. 2, 350; Vol. 3, 168ff
 +
|-
 +
|}

Revision as of 22:50, 17 February 2012

2020 Mathematics Subject Classification: Primary: 12Fxx [MSN][ZBL]

A transcendental extension of a field $k$ is a field extension that is not algebraic (cf. Extension of a field). An extension $K/k$ is transcendental if and only if the field $K$ contains elements that are transcendental over $k$, that is, elements that are not roots of any non-zero polynomial with coefficients in $k$.

The elements of a set $X\subset K$ are called algebraically independent over $k$ if for any finite set $x_1,\dots,x_m \in X$ and any non-zero polynomial $F(X_1,\dots,X_m)$ with coefficients in $k$, $$F(x_1,\dots,x_m)\ne 0.$$ The elements of $X$ are transcendental over $k$. If $X\subset K$ is a maximal set of algebraically independent elements over $k$, then $X$ is called a transcendence basis of $K$ over $k$. The cardinality of $X$ is called the transcendence degree of $K$ over $k$ and is an invariant of the extension $K/k$. For a tower of fields $L\supset K\supset k$, the transcendence degree of $L/k$ is equal to the sum of the transcendence degrees of $L/K$ and $K/k$.

If all elements of a set $X$ are algebraically independent over $k$, then the extension $k(X)$ is called purely transcendental. In this case the field $k(X)$ is isomorphic to the field of rational functions in the set of variables $X$ over $k$. Any field extension $L/k$ can be represented as a tower of extensions $L\supset K\supset k$, where $L/K$ is an algebraic and $K/k$ is a purely transcendental extension. If $K$ can be chosen so that $L/K$ is a separable extension, then the extension $K/k$ is called separably generated, and the transcendence basis of $K$ over $k$ is called a separating basis. If $L$ is separably generated over $k$, then $L$ is separable over $k$. In the case when the extension $L/k$ is finitely generated, the converse holds as well. By definition, an extension $K/k$ is separable if and only if any derivation (cf. Derivation in a ring) of $k$ extends to $K$. Such an extension is uniquely determined for any derivation if and only if the extension $K/k$ is algebraic.

References

[Bo] N. Bourbaki, "Algebra", Elements of mathematics, 1, Springer (1988) pp. Chapt. 4–6 (Translated from French)
[ZaSa] O. Zariski, P. Samuel, "Commutative algebra", 1, Springer (1975)



Comments

The Noether normalization lemma says that if $A$ is an integral domain that is finitely generated as a ring over a field $k$, then there exist $x_1,\dots,x_r \in A$ that are algebraically independent over $k$ such that $A$ is integral over $k[x_1,\dots,x_r]$.

References

[Co] P.M. Cohn, "Algebra", 1–2, Wiley (1989) pp. Vol. 2, 350; Vol. 3, 168ff
How to Cite This Entry:
Transcendental extension. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Transcendental_extension&oldid=21155
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article