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]

One can then check that the set of all polynomials with coefficients in the ring R, together with the addition $+$ and the multiplication $\cdot$ mentioned above, forms itself a ring, the polynomial ring over R, which is denoted by R[X].

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

$\sum _{i=0}^{n}a_{i}X^{i}+\sum _{i=0}^{n}b_{i}X^{i}=\sum _{i=0}^{n}(a_{i}+b_{i})X^{i}$

and

$\left(\sum _{i=0}^{n}a_{i}X^{i}\right)\left(\sum _{j=0}^{m}b_{j}X^{j}\right)=\sum _{k=0}^{m+n}\left(\sum _{\mu +\nu =k}a_{\mu }b_{\nu }\right)X^{k}.$

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

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.

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

Polynomials in n variables 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∈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 $(a_{0},a_{1},...,a_{n},0,0,...)$ (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∈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∈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 Y^j Z^k), 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.