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''polynomial ring''
 
''polynomial ring''
  
A [[Ring|ring]] whose elements are polynomials (cf. [[Polynomial|Polynomial]]) with coefficients in some fixed [[Field|field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r0824101.png" />. Rings of polynomials over an arbitrary commutative associative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r0824102.png" />, for example, over the ring of integers, are also discussed. The accepted notation for the ring of polynomials in a finite set of variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r0824103.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r0824104.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r0824105.png" />. It is possible to speak of a ring of polynomials in an infinite set of variables if it is assumed that each individual polynomial depends only on a finite number of variables. A ring of polynomials over a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r0824106.png" /> is a (commutative) [[Free algebra|free algebra]] with an identity over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r0824107.png" />; the set of variables serves as a system of free generators of this algebra.
+
A [[ring]] whose elements are [[polynomial]]s with coefficients in some fixed [[field]] $k$. Rings of polynomials over an arbitrary commutative associative ring $R$, for example, over the ring of integers, are also discussed. The accepted notation for the ring of polynomials in a finite set of variables $x_1,\ldots,x_n$ over $R$ is $R[x_1,\ldots,x_n]$. It is possible to speak of a ring of polynomials in an infinite set of variables if it is assumed that each individual polynomial depends only on a finite number of variables. A ring of polynomials over a ring $R$ is a (commutative) [[free algebra]] with an identity over $R$; the set of variables serves as a system of free generators of this algebra.
  
A ring of polynomials over an arbitrary [[Integral domain|integral domain]] is itself an integral domain. A ring of polynomials over a [[Factorial ring|factorial ring]] is itself factorial.
+
A ring of polynomials over an arbitrary [[integral domain]] is itself an integral domain. A ring of polynomials over a [[factorial ring]] is itself factorial.
  
For a ring of polynomials in a finite number of variables over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r0824108.png" /> there is Hilbert's basis theorem: Every ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r0824109.png" /> is finitely generated (as an ideal) (cf. [[Hilbert theorem|Hilbert theorem]]). A ring of polynomials in one variable over a field, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241010.png" /> is a [[Principal ideal ring|principal ideal ring]], that is, each ideal of it is generated by one element. Moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241011.png" /> is a [[Euclidean ring|Euclidean ring]]. This property of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241012.png" /> gives one the possibility of comprehensively describing the finitely-generated modules over it and, in particular, of reducing linear operators in a finite-dimensional vector space to canonical form (see [[Jordan matrix|Jordan matrix]]). For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241013.png" /> the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241014.png" /> is not a principal ideal ring.
+
For a ring of polynomials in a finite number of variables over a field $k$ there is Hilbert's basis theorem: Every ideal in $k[x_1,\ldots,x_n]$ is finitely generated (as an ideal) (cf. [[Hilbert theorem]]). A ring of polynomials in one variable over a field, $k[x]$ is a [[principal ideal ring]], that is, each ideal of it is generated by one element. Moreover, $k[x]$ is a [[Euclidean ring]]. This property of $k[x]$ gives one the possibility of comprehensively describing the finitely-generated modules over it and, in particular, of reducing linear operators in a finite-dimensional vector space to canonical form (see [[Jordan matrix]]). For $n>1$ the ring $k[x_1,\ldots,x_n]$ is not a principal ideal ring.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241015.png" /> be a commutative associative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241016.png" />-algebra with an identity, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241017.png" /> be an element of the Cartesian power <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241018.png" />. Then there is a unique <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241019.png" />-algebra homomorphism of the ring of polynomials in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241020.png" /> variables into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241021.png" />,
+
Let $S$ be a commutative associative $k$-algebra with an identity, and let $a = (a_1,\ldots,a_n)$ be an element of the Cartesian power $S^n$. Then there is a unique $k$-algebra homomorphism of the ring of polynomials in $n$ variables into $S$,
 +
$$
 +
\phi_a : k[x_1,\ldots,x_n] \rightarrow S
 +
$$
 +
for which $\phi(x_i) = a_i$, for all $i = 1,\ldots,n$, and $\phi_a(1)$ is the identity of $S$. The image of a polynomial $f \in k[x_1,\ldots,x_n]$ under this homomorphism is called its value at the point $a$. A point $a \in S^n$ is called a zero of a system of polynomials $F \subset k[x_1,\ldots,x_n]$ if the value of each polynomial from $F$ at this point is $0$. For a ring of polynomials there is Hilbert's [[Nullstellen Satz]]: Let $\mathfrak{A}$ be an ideal in the ring $R = k[x_1,\ldots,x_n]$, let $M$ be the set of zeros of $\mathfrak{A}$ in $\bar k^n$, where $\bar k$ is the algebraic closure of $k$, and let $g$ be a polynomial in $R$ vanishing at all points of $M$. Then there is a natural number $m$ such that $g^m \in \mathfrak{A}$ (cf. [[Hilbert theorem]]).
  
<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/r/r082/r082410/r08241022.png" /></td> </tr></table>
+
Let $A$ be an arbitrary module over the ring $R = k[x_1,\ldots,x_n]$. Then there are free $R$-modules $X_0,\ldots,X_n$ and homomorphisms $X_i \rightarrow x_{i-1}$ such that the sequence of homomorphisms
 
+
$$
for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241023.png" />, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241024.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241025.png" /> is the identity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241026.png" />. The image of a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241027.png" /> under this homomorphism is called its value at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241028.png" />. A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241029.png" /> is called a zero of a system of polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241030.png" /> if the value of each polynomial from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241031.png" /> at this point is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241032.png" />. For a ring of polynomials there is Hilbert's Nullstellen Satz: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241033.png" /> be an ideal in the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241034.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241035.png" /> be the set of zeros of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241036.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241037.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241038.png" /> is the algebraic closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241039.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241040.png" /> be a polynomial in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241041.png" /> vanishing at all points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241042.png" />. Then there is a natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241043.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241044.png" /> (cf. [[Hilbert theorem|Hilbert theorem]]).
+
0 \leftarrow A \leftarrow X_0 \leftarrow \cdots \leftarrow X_n \leftarrow 0
 
+
$$
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241045.png" /> be an arbitrary module over the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241046.png" />. Then there are free <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241047.png" />-modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241048.png" /> and homomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241049.png" /> such that the sequence of homomorphisms
+
is exact, that is, the kernel of one homomorphism is the image of the next. This result is one possible formulation of the [[Hilbert theorem]] on [[Syzygy|syzygies]] for a ring of polynomials.
 
 
<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/r/r082/r082410/r08241050.png" /></td> </tr></table>
 
 
 
is exact, that is, the kernel of one homomorphism is the image of the next. This result is one possible formulation of the [[Hilbert theorem|Hilbert theorem]] on syzygies for a ring of polynomials.
 
  
 
A finitely-generated projective module over a ring of polynomials in a finite number of variables with coefficients from a principal ideal ring is free (see [[#References|[5]]], [[#References|[6]]]); this is the solution of Serre's problem.
 
A finitely-generated projective module over a ring of polynomials in a finite number of variables with coefficients from a principal ideal ring is free (see [[#References|[5]]], [[#References|[6]]]); this is the solution of Serre's problem.
  
Only in certain particular cases are there answers to the following questions: 1) Is the group of automorphisms of a ring of polynomials generated by elementary automorphisms? 2) Is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241051.png" /> generated by some set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241052.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241053.png" /> is a non-zero constant? 3) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241054.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241055.png" />, must <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241056.png" /> be isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082410/r08241057.png" />?
+
Only in certain particular cases are there answers to the following questions: 1) Is the group of automorphisms of a ring of polynomials generated by elementary automorphisms? 2) Is $k[x_1,\ldots,x_n]$ generated by some set $f_1,\ldots,f_n$ for which $\det|\partial f_i / \partial x_j|$ is a non-zero constant? 3) If $S \otimes k[y]$ is isomorphic to $k[x_1,\ldots,x_n]$, must $S$ be isomorphic to $k[x_1,\ldots,x_{n-1}]$?
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Lang,  "Algebra" , Addison-Wesley  (1974)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Bourbaki,  "Algèbre" , ''Eléments de mathématiques'' , '''2''' , Masson  (1981)  pp. Chapts. 4; 5; 6</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  D. Hilbert,  "Ueber die vollen Invariantensysteme"  ''Math. Ann.'' , '''42'''  (1893)  pp. 313–373</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  D. Hilbert,  "Ueber die Theorie der algebraischen Formen"  ''Math. Ann.'' , '''36'''  (1890)  pp. 473–534</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.A. Suslin,  "Projective modules over a polynomial ring are free"  ''Soviet Math. Dokl.'' , '''17''' :  4  (1976)  pp. 1160–1164  ''Dokl. Akad. Nauk SSSR'' , '''229'''  (1976)  pp. 1063–1066</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  D. Quillen,  "Projective modules over polynomial rings"  ''Invent. Math.'' , '''36'''  (1976)  pp. 167–171</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  S. Lang,  "Algebra" , Addison-Wesley  (1974)</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  N. Bourbaki,  "Algèbre" , ''Eléments de mathématiques'' , '''2''' , Masson  (1981)  pp. Chapts. 4; 5; 6</TD></TR>
 +
<TR><TD valign="top">[3]</TD> <TD valign="top">  D. Hilbert,  "Ueber die vollen Invariantensysteme"  ''Math. Ann.'' , '''42'''  (1893)  pp. 313–373</TD></TR>
 +
<TR><TD valign="top">[4]</TD> <TD valign="top">  D. Hilbert,  "Ueber die Theorie der algebraischen Formen"  ''Math. Ann.'' , '''36'''  (1890)  pp. 473–534</TD></TR>
 +
<TR><TD valign="top">[5]</TD> <TD valign="top">  A.A. Suslin,  "Projective modules over a polynomial ring are free"  ''Soviet Math. Dokl.'' , '''17''' :  4  (1976)  pp. 1160–1164  ''Dokl. Akad. Nauk SSSR'' , '''229'''  (1976)  pp. 1063–1066</TD></TR>
 +
<TR><TD valign="top">[6]</TD> <TD valign="top">  D. Quillen,  "Projective modules over polynomial rings"  ''Invent. Math.'' , '''36'''  (1976)  pp. 167–171</TD></TR>
 +
</table>
 +
 
 +
{{TEX|done}}

Revision as of 21:56, 29 December 2015

polynomial ring

A ring whose elements are polynomials with coefficients in some fixed field $k$. Rings of polynomials over an arbitrary commutative associative ring $R$, for example, over the ring of integers, are also discussed. The accepted notation for the ring of polynomials in a finite set of variables $x_1,\ldots,x_n$ over $R$ is $R[x_1,\ldots,x_n]$. It is possible to speak of a ring of polynomials in an infinite set of variables if it is assumed that each individual polynomial depends only on a finite number of variables. A ring of polynomials over a ring $R$ is a (commutative) free algebra with an identity over $R$; the set of variables serves as a system of free generators of this algebra.

A ring of polynomials over an arbitrary integral domain is itself an integral domain. A ring of polynomials over a factorial ring is itself factorial.

For a ring of polynomials in a finite number of variables over a field $k$ there is Hilbert's basis theorem: Every ideal in $k[x_1,\ldots,x_n]$ is finitely generated (as an ideal) (cf. Hilbert theorem). A ring of polynomials in one variable over a field, $k[x]$ is a principal ideal ring, that is, each ideal of it is generated by one element. Moreover, $k[x]$ is a Euclidean ring. This property of $k[x]$ gives one the possibility of comprehensively describing the finitely-generated modules over it and, in particular, of reducing linear operators in a finite-dimensional vector space to canonical form (see Jordan matrix). For $n>1$ the ring $k[x_1,\ldots,x_n]$ is not a principal ideal ring.

Let $S$ be a commutative associative $k$-algebra with an identity, and let $a = (a_1,\ldots,a_n)$ be an element of the Cartesian power $S^n$. Then there is a unique $k$-algebra homomorphism of the ring of polynomials in $n$ variables into $S$, $$ \phi_a : k[x_1,\ldots,x_n] \rightarrow S $$ for which $\phi(x_i) = a_i$, for all $i = 1,\ldots,n$, and $\phi_a(1)$ is the identity of $S$. The image of a polynomial $f \in k[x_1,\ldots,x_n]$ under this homomorphism is called its value at the point $a$. A point $a \in S^n$ is called a zero of a system of polynomials $F \subset k[x_1,\ldots,x_n]$ if the value of each polynomial from $F$ at this point is $0$. For a ring of polynomials there is Hilbert's Nullstellen Satz: Let $\mathfrak{A}$ be an ideal in the ring $R = k[x_1,\ldots,x_n]$, let $M$ be the set of zeros of $\mathfrak{A}$ in $\bar k^n$, where $\bar k$ is the algebraic closure of $k$, and let $g$ be a polynomial in $R$ vanishing at all points of $M$. Then there is a natural number $m$ such that $g^m \in \mathfrak{A}$ (cf. Hilbert theorem).

Let $A$ be an arbitrary module over the ring $R = k[x_1,\ldots,x_n]$. Then there are free $R$-modules $X_0,\ldots,X_n$ and homomorphisms $X_i \rightarrow x_{i-1}$ such that the sequence of homomorphisms $$ 0 \leftarrow A \leftarrow X_0 \leftarrow \cdots \leftarrow X_n \leftarrow 0 $$ is exact, that is, the kernel of one homomorphism is the image of the next. This result is one possible formulation of the Hilbert theorem on syzygies for a ring of polynomials.

A finitely-generated projective module over a ring of polynomials in a finite number of variables with coefficients from a principal ideal ring is free (see [5], [6]); this is the solution of Serre's problem.

Only in certain particular cases are there answers to the following questions: 1) Is the group of automorphisms of a ring of polynomials generated by elementary automorphisms? 2) Is $k[x_1,\ldots,x_n]$ generated by some set $f_1,\ldots,f_n$ for which $\det|\partial f_i / \partial x_j|$ is a non-zero constant? 3) If $S \otimes k[y]$ is isomorphic to $k[x_1,\ldots,x_n]$, must $S$ be isomorphic to $k[x_1,\ldots,x_{n-1}]$?

References

[1] S. Lang, "Algebra" , Addison-Wesley (1974)
[2] N. Bourbaki, "Algèbre" , Eléments de mathématiques , 2 , Masson (1981) pp. Chapts. 4; 5; 6
[3] D. Hilbert, "Ueber die vollen Invariantensysteme" Math. Ann. , 42 (1893) pp. 313–373
[4] D. Hilbert, "Ueber die Theorie der algebraischen Formen" Math. Ann. , 36 (1890) pp. 473–534
[5] A.A. Suslin, "Projective modules over a polynomial ring are free" Soviet Math. Dokl. , 17 : 4 (1976) pp. 1160–1164 Dokl. Akad. Nauk SSSR , 229 (1976) pp. 1063–1066
[6] D. Quillen, "Projective modules over polynomial rings" Invent. Math. , 36 (1976) pp. 167–171
How to Cite This Entry:
Ring of polynomials. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ring_of_polynomials&oldid=14262
This article was adapted from an original article by Yu.A. Bakhturin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article