Polynomial 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 than 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
:
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
:
for all elements a of the ring R and
:
for all natural numbers k and l.

The polynomial ring

One can then check that the set of all polynomials with coefficients
in the ring R forms itself a ring, the polynomial ring over R, which is denoted by R[X]. 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
:
is thought of as the polynomial
:
in Y with coefficients in R[X].

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

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].
• 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.
  

  Some uses of polynomial rings

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

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.