Polynomial ring

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

Polynomial rings occur in many parts of mathematics, and the study of their properties was among the main motivations for the development of commutative algebra and ring theory. Polynomial rings and their ideals are fundamental in algebraic geometry. Many classes of rings, such as unique factorization domains, regular rings, group rings, rings of formal power series, Ore polynomials, graded rings, are generalizations of polynomial rings.

A closely related notion is that of the ring of polynomial functions on a vector space, and, more generally, ring of regular functions on an algebraic variety.

The polynomial ring K[X]

Definition

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

${\displaystyle p=p_{0}+p_{1}X+p_{2}X^{2}+\cdots +p_{m-1}X^{m-1}+p_{m}X^{m},}$

where p₀, p₁, ..., p_m, the coefficients of p, are elements of K, and X, X², ..., are symbols, which are considered as "powers" of X, and, by convention, follow the usual rules of exponentiation: X⁰ = 1, X¹ = X, and ${\displaystyle X^{k}\,X^{l}=X^{k+l}}$ for any nonnegative integers 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^k are equal.

This terminology is suggested by real or complex polynomial functions. However, in general, X and its powers, X^k, 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 scalar multiplication that make it a commutative algebra. These operations are defined according to the ordinary rules for manipulating algebraic expressions. Specifically, if

${\displaystyle p=p_{0}+p_{1}X+p_{2}X^{2}+\cdots +p_{m}X^{m},}$

and

${\displaystyle q=q_{0}+q_{1}X+q_{2}X^{2}+\cdots +q_{n}X^{n},}$

then

${\displaystyle p+q=r_{0}+r_{1}X+r_{2}X^{2}+\cdots +r_{k}X^{k},}$

and

${\displaystyle pq=s_{0}+s_{1}X+s_{2}X^{2}+\cdots +s_{l}X^{l},}$

where k = max(m, n), l = m + n,

${\displaystyle r_{i}=p_{i}+q_{i}}$

and

${\displaystyle s_{i}=p_{0}q_{i}+p_{1}q_{i-1}+\cdots +p_{i}q_{0}.}$

The sequence of coefficients in the product of two polynomials is the discrete convolution (or Cauchy product) 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 = p₀ is reduced to its term which is independent of X, that is

${\displaystyle p_{0}(q_{0}+q_{1}X+\dots q_{n}X^{n})=p_{0}q_{0}+(p_{0}q_{1})X+\cdots +(p_{0}q_{n})X^{n}}$

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 sequence of elements of K, (p₀, p₁, p₂, ... ) 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 p_n = 0 for n > m. In this case, the expression

${\displaystyle p_{0}+p_{1}X+p_{2}X^{2}+\cdots +p_{m}X^{m}}$

is considered an alternate notation for the sequence (p₀, p₁, p₂, ..., p_m, 0, 0, ...). 

More generally, the field K can be replaced by any commutative ring 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^k is not zero. In this case the coefficient p_k 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 −1, or defined to be a special symbol −∞.

If K is a field, or more generally an integral domain, then from the definition of multiplication,

${\displaystyle \operatorname {deg} (pq)=\operatorname {deg} (p)+\operatorname {deg} (q).}$

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

Properties of K[X]

Factorization in K[X]

The next property of the polynomial ring is much deeper. Already Euclid noted that every positive integer can be uniquely factored into a product of primes — this statement is now called the fundamental theorem of arithmetic. The proof is based on Euclid's algorithm for finding the greatest common divisor of natural numbers. 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

${\displaystyle p=uq+r,}$

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 polynomial long division. 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 Euclidean rings. Rings for which there exists unique (in an appropriate sense) factorization of nonzero elements into irreducible factors are called unique factorization domains 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 factorization algorithms. However, even the fastest computers are unable to factor large integers with only a few large prime factors. This is the basis of the RSA cryptosystem, widely used for secure Internet communications. For polynomials over the integers, over the rational numbers, or over a finite field, there are efficient algorithms that are implemented in computer algebra systems (see Factorization of polynomials). 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 ${\displaystyle X^{p}-a}$ is irreducible or is a product of polynomials of lower degree.

Another corollary of polynomial division with remainder is the fact that every proper ideal I of K[X] is principal – that is, I consists of the multiples of a single polynomial f. Thus the polynomial ring K[X] is a principal ideal domain, 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 field, 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 field extensions of K. 

Consider an element θ in a commutative ring L that contains K. There is a unique ring homomorphism φ from K[X] into L that maps X to θ and does not affect the elements of K itself (it is the identity map 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.

${\displaystyle \varphi (a_{m}X^{m}+a_{m-1}X^{m-1}+\cdots +a_{1}X+a_{0})=a_{m}\theta ^{m}+a_{m-1}\theta ^{m-1}+\cdots +a_{1}\theta +a_{0}.}$

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, ..., a_m of K. Therefore, φ is surjective and L is a homomorphic image of K[X]. More formally, let Ker φ be the kernel of φ. It is an ideal of K[X] and by the first isomorphism theorem for rings, L is isomorphic 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 principal. That is, there exists a polynomial p ∈ K[X] such that

${\displaystyle L\simeq K[X]/(p),}$

where ${\displaystyle (p)}$ denotes the ideal generated by ${\displaystyle p.}$

A particularly important application is to the case when the larger ring L is a field. Then the polynomial p must be irreducible. Conversely, the primitive element theorem 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 complex numbers is an extension of the field R of real numbers generated by a single element i such that i² + 1 = 0. Accordingly, the polynomial X² + 1 is irreducible over R and

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

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:

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

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 universal property of the ring of polynomials in one variable and explains the ubiquity of polynomial rings in various questions and constructions of ring theory and commutative algebra. 

Modules

The structure theorem for finitely generated modules over a principal ideal domain applies to K[X]. This means that every finitely generated module over K[X] may be decomposed into a direct sum of a free module and finitely many modules of the form ${\displaystyle K[X]/\langle P^{k}\rangle }$, where P is an irreducible polynomial over K and k a positive integer.

Polynomial evaluation

Let K be a field or, more generally, a commutative ring, 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 P(a). 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

${\displaystyle P=X^{2}-1,}$

we have

${\displaystyle P(3)=3^{2}-1=8,}$

${\displaystyle P(X^{2}+1)=(X^{2}+1)^{2}-1=X^{4}+2X^{2}}$

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

${\displaystyle P=P(X),}$

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 ${\displaystyle P\mapsto P(a)}$ defines a ring homomorphism from K[X] into R.

The polynomial function defined by a polynomial P is the function from K into K that is defined by ${\displaystyle x\mapsto P(x).}$ 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 X^q-X both define the zero function. 

The polynomial ring in several variables

Polynomials

A polynomial in n variables X₁, …, X_n with coefficients in a field K is defined analogously to a polynomial in one variable, but the notation is more cumbersome. For any multi-index α = (α₁, …, α_n), where each α_i is a non-negative integer, let

${\displaystyle X^{\alpha }=\prod _{i=1}^{n}X_{i}^{\alpha _{i}}=X_{1}^{\alpha _{1}}\ldots X_{n}^{\alpha _{n}}.}$

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

${\displaystyle p=\sum _{\alpha }p_{\alpha }X^{\alpha },}$

where ${\displaystyle p_{\alpha }=p_{\alpha _{1},\ldots ,\alpha _{n}}\in {K},}$ and only finitely many coefficients p_α are different from 0. The degree of a monomial X^α, frequently denoted |α|, is defined as

${\displaystyle |\alpha |=\sum _{i=1}^{n}\alpha _{i},\ }$

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, ..., X_n], or sometimes K[X], where X is a symbol representing the full set of variables, X = (X₁, ..., X_n), 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 isomorphic to K[X₁][X₂]. 

Polynomials in several variables play fundamental role in algebraic geometry. Many results in commutative and homological algebra 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 free objects in the category of commutative algebras. 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,…, X_n] and algebraic subsets of Kⁿ originating with David Hilbert is known under the name Nullstellensatz (literally: "zero-locus theorem").

• (Weak form, algebraically closed field of coefficients). Let K be an algebraically closed field. Then every maximal ideal m of K[X₁,…, X_n] has the form

${\displaystyle m=(X_{1}-a_{1},\ldots ,X_{n}-a_{n}),\quad a=(a_{1},\ldots ,a_{n})\in {K}^{n}.}$

• (Weak form, any field of coefficients). Let k be a field, K be an algebraically closed field extension of k, and I be an ideal in the polynomial ring k[X₁,…, X_n]. 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 algebraically closed field extension of k, I be an ideal in the polynomial ring k[X₁,…, X_n],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:

${\displaystyle f^{m}\in I,{\text{ for some }}m\in \mathbb {N} .\,}$

Using the notion of the radical of an ideal, the conclusion says that f belongs to the radical of I. As a corollary of this form of Nullstellensatz, there is a bijective correspondence between the radical ideals of K[X₁,…, X_n] for an algebraically closed field K and the algebraic subsets of the n-dimensional affine space Kⁿ. It arises from the map

${\displaystyle I\mapsto V(I),\quad I\subset K[X_{1},\ldots ,X_{n}],\quad V(I)\subset K^{n}.}$

The prime ideals of the polynomial ring correspond to irreducible subvarieties of Kⁿ.

Properties of the ring extension R ⊂ R[X]

One of the basic techniques in commutative algebra is to relate properties of a ring with properties of its subrings. 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,…, X_n], by induction in n. 

Summary of the results

In the following properties, R is a commutative ring and S = R[X₁,…, X_n] 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,…, X_n. Thus to establish each of the properties below, it is sufficient to consider the case n = 1.

• If R is an integral domain then the same holds for S.
• If R is a unique factorization domain then the same holds for S. The proof is based on the Gauss lemma.
• Hilbert's basis theorem: If R is a Noetherian ring, then the same holds for S.
• Suppose that R is a Noetherian ring of finite global dimension. Then

${\displaystyle \operatorname {gl} \,\dim R[X_{1},\ldots ,X_{n}]=\operatorname {gl} \,\dim R+n.}$

An analogous result holds for Krull dimension.

Generalizations

Polynomial rings can be generalized in a great many ways, including polynomial rings with generalized exponents, power series rings, noncommutative polynomial rings, 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 free commutative algebra, the only difference is that it is a free object 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 ring of power series in infinitely many variables. Such a ring is used for constructing the ring of symmetric functions 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: Xⁱ · X^j = X^i+j. A set for which addition makes sense (is closed and associative) is called a monoid. 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 c_n = a_n + b_n 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, c_n is the sum of all a_ib_j 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:

${\displaystyle \sum _{n\in N}a_{n}X^{n}}$

and then the formulas for addition and multiplication are the familiar:

${\displaystyle \left(\sum _{n\in N}a_{n}X^{n}\right)+\left(\sum _{n\in N}b_{n}X^{n}\right)=\sum _{n\in N}(a_{n}+b_{n})X^{n}}$

and

${\displaystyle \left(\sum _{n\in N}a_{n}X^{n}\right)\cdot \left(\sum _{n\in N}b_{n}X^{n}\right)=\sum _{n\in N}\left(\sum _{i+j=n}a_{i}b_{j}\right)X^{n}}$

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

Some authors such as (Lang 2002, II,§3) 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, (Osbourne 2000, §4.4).

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 ring completion 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 monoid ring R[N], where the monoid N is the free monoid 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 > 1. 

Differential and skew-polynomial rings

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

A differential polynomial ring is a ring of differential operators formed from a ring R and a derivation 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 composition of operators is denoted as the usual multiplication. It follows that the relation δ(ab) = aδ(b) + δ(a)b may be rewritten as

${\displaystyle X\cdot a=a\cdot X+\delta (a).}$

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 Weyl algebra, takes R to be a (usual) polynomial ring k[Y], and δ to be the standard polynomial derivative ${\displaystyle {\tfrac {\partial }{\partial Y}}}$. Taking a =Y in the above relation, one gets the canonical commutation relation, X·Y − Y·X = 1. Extending this relation by associativity and distributivity allows explicitly constructing the Weyl algebra.(Lam 2001, §1,ex1.9). 

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.(Lam 2001, §1,ex 1.11) Skew polynomial rings are closely related to crossed product algebras.