Polynomial ring

In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring 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 have influenced much of mathematics, from the Hilbert basis theorem, to the construction of splitting fields, and to the understanding of a linear operator. Many important conjectures involving polynomial rings, such as Serre's problem, have influenced the study of other rings, and have influenced even the definition of other rings, such as group rings and rings of formal power series.

A closely related notion is that of the ring of polynomial functions on a vector space.

The polynomial ring K[X]

Definition

The polynomial ring, K[X], in X over a field K is defined[1] 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 p0, p1,…, pm, the coefficients of p, are elements of K, and X, X 2, are formal symbols ("the powers of X"). By convention, X 0 = 1, X 1 = X, and the product of the powers of X is defined by the familiar formula

X^k\, X^l = X^{k+l},

where k and l are any natural numbers. The symbol X is called an indeterminate[2] or variable.[3]

Two polynomials are defined to be equal if and only if the corresponding coefficients for each power of X are equal, however terms with zero coefficient, 0X k, may be added or omitted.

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 requiring that X commute with all elements of K.

Polynomials in X are added and multiplied according to the ordinary rules for manipulating algebraic expressions, creating the structure of a ring. 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

r_i=p_i+q_i

and

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

If necessary, the polynomials p and q are extended by adding "dummy terms" with zero coefficients, so that the expressions for ri and si are always defined.

A more rigorous, but less intuitive treatment[4] defines a polynomial as an infinite tuple, or ordered 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 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 (p0, p1, p2, … pm, 0, 0, …).

More generally, the field K can be replaced by any commutative ring R when taking 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.[5] In this case the coefficient pk is called the leading coefficient.[6] In the special case of zero polynomial, all of whose coefficients are zero, the degree has been variously left undefined,[7] defined to be −1,[8] or defined to be a special symbol −∞.[9]

If K is a field, or more generally an integral domain, then from the definition of multiplication,[10]

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

It follows immediately that if K is an integral domain then so is K[X].[11]

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, where q ≠ 0, one can write

p = uq + r,\

where the quotient u and the remainder r are polynomials, the degree of r is less than the degree of q, and a decomposition with these properties is unique. The quotient and the remainder are found using the 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 step such 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; all such rings 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.

Another corollary of the polynomial division with the remainder is the fact that every proper ideal I of K[X] is principal, i.e. 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. Also 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 so if and only it 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.

Suppose that a commutative ring L contains K and there exists an element θ of L such that the ring L is generated by θ over K. Thus any element of L is a linear combination of powers of θ with coefficients in K. Then there is a unique ring homomorphism φ from K[X] into L which does not affect the elements of K itself (it is the identity map on K) and maps each power of X to the same power of θ. Its effect on the general polynomial amounts to "replacing X with θ":

\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.

By the assumption, 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 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: there exists a polynomial p∈K[X] such that

L \simeq K[X]/(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

\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 universal property of the ring of polynomials in one variable and explains 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 over 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 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

P=X^2-1,

we have

P(3) = 3^2-1 = 8,
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

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 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 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.

The polynomial ring in several variables

Polynomials

A polynomial in n variables X1,…, Xn 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,…, αn), where each αi is a non-negative integer, let

X^\alpha = \prod_{i=1}^n X_i^{\alpha_i} = X_1^{\alpha_1}\ldots X_n^{\alpha_n}, \quad p_\alpha = p_{\alpha_1\ldots\alpha_n}\in\mathbb{K}.\

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

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

and only finitely many coefficients pα are different from 0. The degree of a monomial Xα, 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 isomorphic to K[X1][X2]. This ring plays fundamental role in algebraic geometry. Many results in commutative and homological algebra originated in the study of its ideals and modules over this ring.

A polynomial ring with coefficients in \mathbb{Z} 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 algebraic subsets of Kn originating with David Hilbert is known under the name Nullstellensatz (literally: "zero-locus theorem").

m = (X_1-a_1, \ldots, X_n-a_n), \quad a = (a_1, \ldots, a_n) \in \mathbb{K}^n.
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,…, Xn] for an algebraically closed field K and the algebraic subsets of the n-dimensional affine space 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 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,…, 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.

\operatorname{gl}\,\dim R[X_1,\ldots,X_n] = \operatorname{gl}\, \dim R + n.
An analogous result holds for Krull dimension.

Generalizations

Polynomial rings have been 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

The possibility to allow an infinite set of indeterminates is not really a generalization, as the ordinary notion of polynomial ring allows for it. It is then still true that each monomial involves only a finite number of indeterminates (so that its degree remains finite), and that each polynomial is a linear combination of monomials, which by definition involves only finitely many of them. This explains why such polynomial rings are relatively seldom considered: each individual polynomial involves only finitely many indeterminates, and even any finite computation involving polynomials remains inside some subring of polynomials in finitely many indeterminates.

In the case of infinitely many indeterminates, one can consider a ring strictly larger than the polynomial ring but smaller than the power series ring, by taking the subring of the latter formed by power series whose monomials have a bounded degree. Its elements still have a finite degree and are therefore are somewhat like polynomials, but it is possible for instance to take the sum of all indeterminates, which is not a polynomial. A ring of this kind plays a role in constructing the ring of symmetric functions.

Generalized exponents

Main article: Monoid ring

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·Xj = Xi+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 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 (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

Main article: Formal 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 be seen as the completion of the polynomial ring.

Noncommutative polynomial rings

Main article: Free algebra

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

Main article: Ore extension

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

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 \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 to construct explicitly 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 to construct a skew-polynomial ring.(Lam 2001, §1,ex 1.11) Skew polynomial rings are closely related to crossed product algebras.

See also

References

  1. ↑ Following Herstein p. 153
  2. ↑ Herstein, Hall p. 73
  3. ↑ Lang p. 97
  4. ↑ Following Hall p.72-73
  5. ↑ Herstein p. 154
  6. ↑ Lang p.100
  7. ↑ Anton, Howard; Bivens, Irl C.; Davis, Stephen (2012), Calculus Single Variable, John Wiley & Sons, p. 31, ISBN 9780470647707.
  8. ↑ Sendra, J. Rafael; Winkler, Franz; Pérez-Diaz, Sonia (2007), Rational Algebraic Curves: A Computer Algebra Approach, Algorithms and Computation in Mathematics 22, Springer, p. 250, ISBN 9783540737247.
  9. ↑ Eves, Howard Whitley (1980), Elementary Matrix Theory, Dover, p. 183, ISBN 9780486150277.
  10. ↑ Herstein p.155, 162
  11. ↑ Herstein p.162