Polynomial ring

In abstract algebra, a polynomial ring is the set of polynomials in one or more variables with coefficients in a ring.

Definition of a polynomial

In real analysis, a polynomial is a certain type of a function of one or several variables (see polynomial), or in other words, a polynomial function.

This definition cannot be adapted to a general ring, however. For example, over the ring Z/2Z of integers modulo 2, the polynomial

P(X)=X²+X=X(X+1)

takes only the value 0, as when k is an integer, k(k+1) is always even. But we would expect P(X) to be different from the zero polynomial.

The approach taken is then the following. Let R be a ring. A polynomial P(X) is defined to be a formal expression of the form

$P(X)=a_{m}X^{m}+a_{m-1}X^{m-1}+\cdots +a_{1}X+a_{0}$

where the coefficients a₀, …, a_m are elements of the ring R, and X is considered to be a formal symbol. Two polynomials are considered to be equal if and only if the corresponding coefficients for each power of X are equal. Polynomials with coefficients in R can be added by simply adding the corresponding coefficients and multiplied using the distributive law, and the rules

$X\,a=a\,X$

for all elements a of the ring R and

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

for all natural numbers k and l.

The polynomial ring R[X]

The set of all polynomials with coefficients in the ring R, together with the addition + and the multiplication · mentioned above, forms itself a ring, the polynomial ring over R, which is denoted by R[X]. One can think of the ring R[X] as arising from R by adding one new element X to R and only requiring that X commute with all elements of R. In order for R[X] to form a ring, all sums of powers of X have to be included as well.

Formally the two ring operations are functions defined on R[X] × R[X] with values in R[X], given by the formulas

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

and

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

where only a finite number of terms are nonzero in these formally infinite sums.

If R is commutative, then R[X] is an algebra over R.

The polynomial ring in several variables

Given two variables X and Y, one constructs the polynomial ring R[X], and then, on top of it, the ring (R[X])[Y]. This ring is considered the polynomial ring in the two variables R[X,Y].

For example, the polynomial

$P(X,Y)=X^{2}Y^{2}+4XY^{2}+5X^{3}-8Y^{2}+6XY-2Y+7$

is thought of as the polynomial

$(X^{2}+4X-8)Y^{2}+(6X-2)Y+(5X^{3}+7)$

in Y with coefficients in R[X].

In similar fashion, the ring R[X₁, …, X_n] in n variables X₁, …, X_n is constructed.

Equivalent definition

For a ring R and the set of nonnegative integers N, the ring of polynomials in n variables over R can also be defined as functions from Nⁿ into R which are zero everywhere except for a finite number of points, with the addition and R-multiplication defined in the canonical way, and multiplication defined by the convolution

$P*Q:k\mapsto \sum _{i+j=k}P_{i}\,Q_{j}~,$

where i,j,k in Nⁿ are the (multi-)indices corresponding to respective powers of the indeterminates (and $P_{i},Q_{j}$ are the associated coefficients of the respective polynomial).

The link with the traditional notation is made by writing as $X_{p}^{q}$ the elements of the canonical basis of this free module, which are the functions associating to a vector (0…0,q,0…0) of Nⁿ the value 1_R, and zero to any other vector of Nⁿ (where q is in the p-th place of the vector).

Understanding this definition

To get a better idea of the meaning of this definition, start by considering the case n=1. It is easily seen that R[X] is nothing else than the set of finite sequences $(a_{0},a_{1},\ldots ,a_{n},0,0,\ldots )$ (finite meaning equal to zero from a certain place onwards, i.e. referring to the number of nonzero elements), with the notation Xⁱ=(0,…,0,1,0,…), the 1 being at the i-th position (starting with i=0, and assuming 1 in R for simplicity). Then the above convolution product reproduces exactly the usual formula X^{i Xj = Xi+j. Such a sequence is nothing else than a function from N to R, with its value at i in N denoted by a_i instead of f(i). Now, polynomials in several (e.g. 3) variables (e.g. X,Y,Z) have coefficients with as many indices as there are variables (e.g. a_i,j,k in this example, for the coefficient of Xi Yj Zk), i.e. they are functions from Nn (here N3 = N×N×N), and it is a straightforward exercise to see that once again the convolution product corresponds to "summing up respective powers of the variables", or more precisely, to adding up coefficients of monomials whose product would yield the given power of the unknowns.}

Properties

• If R is a field, then R[X] is a principal ideal domain (and even a Euclidean domain).
• If R is a unique factorization domain, so is R[X₁, …, X_n]. This follows from Gauss's lemma.
• If R is an integral domain, so is R[X₁, …, X_n].
• If R is Noetherian, then R[X₁, …, X_n] is Noetherian. This is the Hilbert basis theorem.
• Every commutative ring that is a finitely-generated algebra over a field can be written as a quotient of a polynomial ring.
• For any unital R-algebra A, one can canonically associate to every P in R[X] a map $P_{A}:A\to A;x\mapsto P(x)$, where P(0)=a₀ is identified with a₀·1_A.
• For each (fixed) element x of a unital R-algebra A, we have the substitution map $s_{x}:R[X]\to A,P\mapsto P(x)$, which is a morphism of R-algebras.

Some uses of polynomial rings

Factoring out ideals from a polynomial ring is an important tool for constructing new rings out of known ones.

For instance, the clean construction of finite fields involves the use of those operations, starting out with the field of integers modulo some prime number as the coefficient ring R (see modular arithmetic).

The complex number planes can be presented as quotients:

• the field of complex numbers C can be defined as R[X]/(1+X²)
• the ring of split-complex numbers D can be defined as R[X]/(1−X²).
• the ring of dual numbers N can be defined as R[X]/(X²).

An interesting example of a ring obtained by using polynomials is the ring of Frobenius polynomials, where the ring multiplication is given by function composition, rather than by polynomial multiplication.

Polynomial rings can be used to classify simple field extensions.

Generalizations

Polynomial rings have be generalized in a great many ways, including polynomial rings with generalized exponents, power series rings, non-commutative polynomial rings, and skew-polynomial rings.

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·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:

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

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

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

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

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.

Non-commutative 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 non-commuting 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 length n, with multiplication given by concatenation.

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 non-commutative polynomial ring in n variables with coefficients in the commutative ring R is the free associative, unital (but not necessarily commutative) R-algebra.

Differential and skew-polynomial rings

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

Main article: Weyl algebra

A differential polynomial ring is formed from a ring R and a derivation δ of R into R. Then the multiplication is extended from the relation X·a = a·X + δ(a). The standard example, called a Weyl algebra, takes R to be a polynomial ring k[t], and X to be the standard polynomial derivative ${\tfrac {\partial }{\partial t}}$. One views the elements of R[X] as differential operators on the polynomial ring k[t], with elements f(t) of R=k[t] acting as multiplication, and X acting as the derivative in t. Labelling t = Y, one gets the canonical commutation relation, X·Y - Y·X = 1, making the ring explicitly a Weyl algebra. This is a fundamentally important ring, (Lam 2001, §1,ex1.9).

The skew-polynomial ring is defined for a ring R and a ring endomorphism f of R, multiplication is extended from the relation X·r = f(r)·X to give an associative multiplication that distributes over the standard addition. More generally, one has a homomorphism F from the monoid N into the endomorphism ring of R, and Xⁿ·r = F(n)(r)·Xⁿ, as in (Lam 2001, §1,ex 1.11). Skew polynomial rings are closely related to crossed product algebras.

References

• Lam, Tsit-Yuen (2001), A First Course in Noncommutative Rings, Berlin, New York: Springer-Verlag, ISBN 978-0-387-95325-0
• Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol. 211, Berlin, New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR1878556
• Osborne, M. Scott (2000), Basic homological algebra, Graduate Texts in Mathematics, vol. 196, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98934-1, MR1757274