# Ring of polynomials

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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 , ); 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 $k[x_1,\ldots,x_{n-1}]$?

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How to Cite This Entry:
Ring of polynomials. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ring_of_polynomials&oldid=37141
This article was adapted from an original article by Yu.A. Bakhturin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article