Polynomial ring

In, especially in the field of , a polynomial ring or polynomial algebra is a (which is also a ) formed from the  of s in one or more s (traditionally also called ) with coefficients in another , often a.

Polynomial rings occur in many parts of mathematics, and the study of their properties was among the main motivations for the development of and. Polynomial rings and their are fundamental in. Many classes of rings, such as s, s, s,, s, s, are generalizations of polynomial rings.

A closely related notion is that of the on a, and, more generally,  on an.

Definition
The polynomial ring, $K[X]$, in $X$ over a $K$ is defined as the set of expressions, called polynomials in $X$, of the form


 * $$p = p_0 + p_1 X + p_2 X^2 + \cdots + p_{m - 1} X^{m - 1} + p_m X^m,$$

where $p_{0}, p_{1}, ..., p_{m}$, the coefficients of $p$, are elements of $K$, and $X, X, ...$, are symbols, which are considered as "powers" of $X$, and, by convention, follow the usual rules of : $X = 1$, $X = X$, and $$ X^k\, X^l = X^{k+l}$$ for any s $k$ and $l$. The symbol $X$ is called an indeterminate or variable.

Two polynomials are defined to be equal when the corresponding coefficients of each $X$ are equal.

This terminology is suggested by or  polynomial functions. However, in general, $X$ and its powers, $X$, are treated as formal symbols, not as elements of the field $K$ or functions over it. One can think of the ring $K[X]$ as arising from $K$ by adding one new element $X$ that is external to $K$ and commute with all elements of $K$.

The polynomial ring in $X$ over $K$ is equipped with an addition, a multiplication and a that make it a. These operations are defined according to the ordinary rules for manipulating algebraic expressions. Specifically, if
 * $$p = p_0 + p_1 X + p_2 X^2 + \cdots + p_m X^m,$$

and
 * $$q = q_0 + q_1 X + q_2 X^2 + \cdots + q_n X^n,$$

then
 * $$p + q = r_0 + r_1 X + r_2 X^2 + \cdots + r_k X^k,$$

and
 * $$pq = s_0 + s_1 X + s_2 X^2 + \cdots + s_l X^l,$$

where $k = max(m, n), l = m + n$,
 * $$r_i=p_i+q_i$$

and
 * $$s_i=p_0 q_i + p_1 q_{i-1} + \cdots + p_i q_0.$$

The of coefficients in the product of two polynomials is the  (or ) of the sequences of coefficients of the polynomials being multiplied.

If necessary, the polynomials $p$ and $q$ are extended by adding "dummy terms" with zero coefficients, so that the expressions for $r_{i}$ and $s_{i}$ are always defined. Specifically, if $m < n$, then $p_{i} = 0$ for $m < i ≤ n$.

The scalar multiplication is the special case of the multiplication where $p = p_{0}$ is reduced to its term which is independent of $X$, that is
 * $$p_0(q_0 + q_1 X + \dots q_nX^n) = p_0q_0 +(p_0q_1)X + \cdots + (p_0q_n)X^n$$

It is straightforward to verify that these three operations satisfy the axioms of a commutative algebra. Therefore, polynomial rings are also called polynomial algebras.

Another equivalent definition is often preferred, although less intuitive, because it is easier to make it completely rigorous, which consists in defining a polynomial as an infinite of elements of $K$, $(p_{0}, p_{1}, p_{2}, ... )$ having the property that only a finite number of the elements are nonzero, or equivalently, a sequence for which there is some $m$ so that pn = 0 for $n > m$. In this case, the expression
 * $$p_0 + p_1 X + p_2 X^2 + \cdots + p_m X^m$$

is considered an alternate notation for the sequence $(p_{0}, p_{1}, p_{2}, ..., p_{m}, 0, 0, ...)$.

More generally, the field $K$ can be replaced by any $R$ for the same construction as above, giving rise to the polynomial ring over $R$, which is denoted $R[X]$.

Degree of a polynomial
The degree of a polynomial p, written deg(p) is the largest k such that the coefficient of X is not zero. In this case the coefficient pk is called the leading coefficient. In the special case of zero polynomial, all of whose coefficients are zero, the degree has been variously left undefined, defined to be &minus;1, or defined to be a special symbol &minus;∞.

If K is a field, or more generally an, then from the definition of multiplication,
 * $$\operatorname{deg}(pq) = \operatorname{deg}(p) + \operatorname{deg}(q).$$

It follows immediately that if K is an integral domain then so is K[X].

Factorization in K[X]
The next property of the polynomial ring is much deeper. Already noted that every positive integer can be uniquely factored into a product of  — this statement is now called the. The proof is based on for finding the  of s. At each step of this algorithm, a pair (a, b), a > b, of natural numbers is replaced by a new pair (b, r), where r is the remainder from the division of a by b, and the new numbers are smaller. Gauss remarked that the procedure of division with the remainder can also be defined for polynomials: given two polynomials p and q with q ≠ 0, one can write


 * $$ p = uq + r,$$

where the quotient u and the remainder r are polynomials, and the degree of r is strictly less than the degree of q. Moreover, a decomposition of this form is unique. The quotient and the remainder are found using. The degree of the polynomial now plays a role similar to the absolute value of an integer; it is strictly less in the remainder r than it is in q, and when repeating this procedure such a decrease cannot go on indefinitely. Therefore, eventually some division will be exact, at which point the last non-zero remainder is the greatest common divisor of the initial two polynomials. Using the existence of greatest common divisors, Gauss was able to simultaneously rigorously prove the fundamental theorem of arithmetic for integers and its generalization to polynomials. In fact there exist other commutative rings than Z and K[X] that similarly admit an analogue of the Euclidean algorithm; rings of this kind are called s. Rings for which there exists unique (in an appropriate sense) factorization of nonzero elements into factors are called s or factorial rings. The given construction shows that all Euclidean rings, and in particular Z and K[X], are unique factorization domains.

Although the Euclidean algorithm allows proving the unique factorization property, it does not provide an algorithm for computing the factorization. For integers, there are. However, even the fastest computers are unable to factor large integers with only a few large prime factors. This is the basis of the, widely used for secure Internet communications. For polynomials over the integers, over the s, or over a, there are efficient algorithms that are implemented in s (see ). On the other hand, there is an example of a field F such that there exist algorithms for the operations of F, but there cannot exist any algorithm for deciding whether a polynomial of the form $$X^p-a$$ is or is a product of polynomials of lower degree.

Another corollary of polynomial division with remainder is the fact that every proper I of K[X] is  – that is, I consists of the multiples of a single polynomial f. Thus the polynomial ring K[X] is a, and for the same reason, every Euclidean domain is a principal ideal domain. Moreover, every principal ideal domain is a unique factorization domain. These deductions make essential use of the fact that the polynomial coefficients lie in a, namely in the polynomial division step, which requires the leading coefficient of q, which is only known to be non-zero, to have an inverse. If R is an integral domain that is not a field then R[X] is neither a Euclidean domain nor a principal ideal domain, however it could still be a unique factorization domain (and will be if and only if R itself is a unique factorization domain, for instance if it is Z or another polynomial ring).

Quotient ring of K[X]
The ring K[X] of polynomials over K is obtained from K by adjoining one element, X. It turns out that any commutative ring L containing K and generated as a ring by a single element in addition to K can be described using K[X]. In particular, this applies to finite s of K.

Consider an element $θ$ in a commutative ring L that contains K. There is a unique $φ$ from K[X] into L that maps X to $θ$ and does not affect the elements of K itself (it is the  on K). This homomorphism is unique since it must map each power of X to the same power of $θ$ and any linear combination of powers of X with coefficients in K to the same linear combination of powers of X. It consists thus of "replacing X with $θ$" in every polynomial.


 * $$ \varphi(a_m X^m + a_{m - 1} X^{m - 1} + \cdots + a_1 X + a_0) =

a_m \theta^m + a_{m - 1} \theta^{m - 1} + \cdots + a_1 \theta + a_0.$$

If L is generated as a ring by adding $θ$ to K, any element of L appears as the right hand side of the last expression for suitable m and elements a0, ..., am of K. Therefore, $φ$ is and L is a homomorphic image of K[X]. More formally, let $Ker φ$ be the of $φ$. It is an of K[X] and by the first  for rings, L is  to the quotient of the polynomial ring K[X] by the ideal $Ker φ$. Since the polynomial ring is a principal ideal domain, this ideal is. That is, there exists a polynomial p ∈ K[X] such that


 * $$ L \simeq K[X]/(p), $$

where $$(p)$$ denotes the ideal generated by $$p.$$

A particularly important application is to the case when the larger ring L is a. Then the polynomial p must be. Conversely, the states that any finite separable field extension L/K can be generated by a single element θ ∈ L and the preceding theory then gives a concrete description of the field L as the quotient of the polynomial ring K[X] by a principal ideal generated by an irreducible polynomial p. As an illustration, the field C of s is an extension of the field R of s generated by a single element i such that i + 1 = 0. Accordingly, the polynomial X + 1 is irreducible over R and


 * $$ \mathbb{C} \simeq \mathbb{R}[X]/(X^2+1). $$

More generally, given a (not necessarily commutative) ring A containing K and an element a of A that commutes with all elements of K, there is a unique ring homomorphism from the polynomial ring K[X] to A that maps X to a:


 * $$ \phi: K[X]\to A, \quad \phi(X)=a.$$

This homomorphism is given by the same formula as before, but it is not surjective in general. The existence and uniqueness of such a homomorphism $φ$ expresses a certain of the ring of polynomials in one variable and explains the ubiquity of polynomial rings in various questions and constructions of  and.

Modules
The applies to K[X]. This means that every finitely generated module over K[X] may be decomposed into a of a  and finitely many modules of the form $$K[X]/\langle P^k \rangle$$, where P is an  over K and k a positive integer.

Polynomial evaluation
Let K be a field or, more generally, a, and R a ring containing K. For any polynomial P in K[X] and any element a in R, the substitution of X by a in P defines an element of R, which is denoted. This element is obtained by, after the substitution, carrying on, in R, the operations indicated by the expression of the polynomial. This computation is called the evaluation of P at a. For example, if we have
 * $$P = X^2 - 1,$$

we have
 * $$\begin{align}

P(3) &= 3^2-1 = 8, \\ P(X^2+1) &= \left(X^2 + 1\right)^2 - 1 = X^4 + 2X^2 \end{align}$$

(in the first example R = K, and in the second one R = K[X]). Substituting X by itself results in
 * $$P = P(X),$$

explaining why the sentences "Let P be a polynomial" and "Let P(X) be a polynomial" are equivalent.

For every a in R, the map $$P \mapsto P(a)$$ defines a from K[X] into R.

The polynomial function defined by a polynomial P is the function from K into K that is defined by $$x\mapsto P(x).$$ If K is an infinite field, two different polynomials define different polynomial functions, but this property is false for finite fields. For example, if K is a field with q elements, then the polynomials 0 and Xq − X both define the zero function.

Polynomials
A polynomial in $n$ indeterminatess $X_{1}, …, X_{n}$ with coefficients in a field $K$ is defined analogously to a polynomial in one indeterminate, but the notation is more cumbersome. For any $α = (α_{1}, …, α_{n})$, where each $α_{i}$ is a non-negative integer, let


 * $$ X^\alpha = \prod_{i=1}^n X_i^{\alpha_i} = X_1^{\alpha_1}\cdots X_n^{\alpha_n}.$$

The product $X^{α}$ is called the of multidegree $α$. A polynomial is a finite linear combination of monomials with coefficients in $K$


 * $$ p = \sum_\alpha p_\alpha X^\alpha,$$

where $$p_\alpha = p_{\alpha_1, \ldots, \alpha_n}\in{K},$$ and only finitely many coefficients $p_{α}$ are different from 0. The degree of a monomial $X^{α}$, frequently denoted $|α|$, is defined as


 * $$ |\alpha| = \sum_{i=1}^n \alpha_i,\ $$

and the degree of a polynomial $p$ is the largest degree of a monomial occurring with non-zero coefficient in the expansion of $p$.

The polynomial ring
Polynomials in n variables with coefficients in K form a commutative ring denoted K[X1, ..., Xn], or sometimes K[X], where X is a symbol representing the full set of variables, X = (X1, ..., Xn), and called the polynomial ring in n variables. The polynomial ring in n variables can be obtained by repeated application of K[X] (the order by which is irrelevant). For example, K[X1, X2] is to K[X1][X2].

Polynomials in several variables play fundamental role in. Many results in and  originated in the study of ideals and modules over polynomial rings.

Polynomial rings may also be referred to as free commutative algebras, since they are the s in the of. Similarly, a polynomial ring with coefficients in the integers is the free commutative ring over its set of variables.

Hilbert's Nullstellensatz
A group of fundamental results concerning the relation between ideals of the polynomial ring K[X1, …, Xn] and of Kn originating with  is known under the name Nullstellensatz (literally: "zero-locus theorem").

Weak form, algebraically closed field of coefficients:Let K be an. Then every m of K[X1, …, Xn] has the form

m = \left(X_1 - a_1,\, \ldots,\, X_n - a_n\right), \quad a = \left(a_1,\, \ldots,\, a_n\right) \in K^n. $ Weak form, any field of coefficients:Let k be a field, K be an of k, and I be an ideal in the polynomial ring k[X1, …, Xn]. Then I contains 1 if and only if the polynomials in I do not have any common zero in Kn. Strong form:Let k be a field, K be an of k, I be an ideal in the polynomial ring k[X1, …,  Xn], and V(I) be the algebraic subset of Kn defined by I. Suppose that f is a polynomial which vanishes at all points of V(I). Then some power of f belongs to the ideal I:
 * $ f^m \in I, \text{ for some } m\in \mathbb{N}. \, $

Using the notion of the, the conclusion says that f belongs to the radical of I. As a corollary of this form of Nullstellensatz, there is a correspondence between the radical ideals of K[X1, …, Xn] for an algebraically closed field K and the algebraic subsets of the n-dimensional  Kn. It arises from the map
 * $ I \mapsto V(I), \quad I\subset K[X_1,\, \ldots,\, X_n], \quad V(I) \subset K^n.$

The s of the polynomial ring correspond to subvarieties of Kn.

Properties of the ring extension R &sub; R[X]
One of the basic techniques in is to relate properties of a ring with properties of its s. The notation R ⊂ S indicates that a ring R is a subring  of a ring S. In this case S is called an overring of R and one speaks of a ring extension. This works particularly well for polynomial rings and allows one to establish many important properties of the ring of polynomials in several variables over a field, K[X1,…, Xn], by induction in n.

Summary of the results
In the following properties, R is a commutative ring and S = R[X1,…, Xn] is the ring of polynomials in n variables over R. The ring extension R ⊂ S can be built from R in n steps, by successively adjoining X1,…, Xn. Thus to establish each of the properties below, it is sufficient to consider the case n = 1.


 * If R is an then the same holds for S.
 * If R is a then the same holds for S. The proof is based on the.
 * : If R is a, then the same holds for S.
 * Suppose that R is a Noetherian ring of finite . Then


 * $$ \operatorname{gl}\,\dim R[X_1,\ldots,X_n] = \operatorname{gl}\, \dim R + n.$$


 * An analogous result holds for.

Generalizations
Polynomial rings can be generalized in a great many ways, including polynomial rings with generalized exponents, power series rings, s, and skew-polynomial rings.

Infinitely many variables
One slight generalization of polynomial rings is to allow for infinitely many indeterminates. Each monomial still involves only a finite number of indeterminates (so that its degree remains finite), and each polynomial is a still a (finite) linear combination of monomials. Thus, any individual polynomial involves only finitely many indeterminates, and any finite computation involving polynomials remains inside some subring of polynomials in finitely many indeterminates. This generalization has the same property of usual polynomial rings, of being the, the only difference is that it is a over an infinite set.

One can also consider a strictly larger ring, by defining as a generalized polynomial an infinite (or finite) formal sum of monomials with a bounded degree. This ring is larger than the usual polynomial ring, as it includes infinite sums of variables. However, it is smaller than the. Such a ring is used for constructing the over an infinite set.

Generalized exponents
A simple generalization only changes the set from which the exponents on the variable are drawn. The formulas for addition and multiplication make sense as long as one can add exponents: X · X = X. A set for which addition makes sense (is closed and associative) is called a. The set of functions from a monoid N to a ring R which are nonzero at only finitely many places can be given the structure of a ring known as R[N], the monoid ring of N with coefficients in R. The addition is defined component-wise, so that if c = a + b, then cn = an + bn for every n in N. The multiplication is defined as the Cauchy product, so that if c = a · b, then for each n in N, cn is the sum of all aibj where i, j range over all pairs of elements of N which sum to n.

When N is commutative, it is convenient to denote the function a in R[N] as the formal sum:
 * $$\sum_{n \in N} a_n X^n$$

and then the formulas for addition and multiplication are the familiar:
 * $$\left(\sum_{n \in N} a_n X^n\right) + \left(\sum_{n \in N} b_n X^n\right) = \sum_{n \in N} (a_n+b_n)X^n$$

and
 * $$\left(\sum_{n \in N} a_n X^n\right) \cdot \left(\sum_{n \in N} b_n X^n\right) = \sum_{n \in N} \left( \sum_{i+j=n} a_ib_j\right)X^n$$

where the latter sum is taken over all i, j in N that sum to n.

Some authors such as go so far as to take this monoid definition as the starting point, and regular single variable polynomials are the special case where N is the monoid of non-negative integers. Polynomials in several variables simply take N to be the direct product of several copies of the monoid of non-negative integers.

Several interesting examples of rings and groups are formed by taking N to be the additive monoid of non-negative rational numbers,. See also.

Power series
Power series generalize the choice of exponent in a different direction by allowing infinitely many nonzero terms. This requires various hypotheses on the monoid N used for the exponents, to ensure that the sums in the Cauchy product are finite sums. Alternatively, a topology can be placed on the ring, and then one restricts to convergent infinite sums. For the standard choice of N, the non-negative integers, there is no trouble, and the ring of formal power series is defined as the set of functions from N to a ring R with addition component-wise, and multiplication given by the Cauchy product. The ring of power series can also be seen as the of the polynomial ring with respect to the ideal generated by $x$.

Noncommutative polynomial rings
For polynomial rings of more than one variable, the products X·Y and Y·X are simply defined to be equal. A more general notion of polynomial ring is obtained when the distinction between these two formal products is maintained. Formally, the polynomial ring in n noncommuting variables with coefficients in the ring R is the R[N], where the monoid N is the  on n letters, also known as the set of all strings over an alphabet of n symbols, with multiplication given by concatenation. Neither the coefficients nor the variables need commute amongst themselves, but the coefficients and variables commute with each other.

Just as the polynomial ring in n variables with coefficients in the commutative ring R is the free commutative R-algebra of rank n, the noncommutative polynomial ring in n variables with coefficients in the commutative ring R is the free associative, unital R-algebra on n generators, which is noncommutative when n &gt; 1.

Differential and skew-polynomial rings
Other generalizations of polynomials are differential and skew-polynomial rings.

A differential polynomial ring is a ring of s formed from a ring R and a δ of R into R. This derivation operates on R, and will be denoted X, when viewed as an operator. The elements of R also operate on R by multiplication. The is denoted as the usual multiplication. It follows that the relation δ(ab) = aδ(b) + δ(a)b may be rewritten as
 * $$X\cdot a = a\cdot X +\delta(a).$$

This relation may be extended to define a skew multiplication between two polynomials in X with coefficients in R, which make them a non-commutative ring.

The standard example, called a, takes R to be a (usual) polynomial ring k[Y], and δ to be the standard polynomial derivative $$\tfrac{\partial}{\partial Y}$$. Taking a =Y in the above relation, one gets the, X·Y &minus; Y·X = 1. Extending this relation by associativity and distributivity allows explicitly constructing the ..

The skew-polynomial ring is defined similarly for a ring R and a ring endomorphism f of R, by extending the multiplication from the relation X·r = f(r)·X to produce an associative multiplication that distributes over the standard addition. More generally, given a homomorphism F from the monoid N of the positive integers into the endomorphism ring of R, the formula Xn·r = F(n)(r)·Xn allows constructing a skew-polynomial ring. Skew polynomial rings are closely related to algebras.