When polynomials are added, subtracted or multiplied it is not terribly important to know what the indeterminant is, but the quotient and remainder operations do require this knowledge, because the division algorithm is stated for polynomials in x. The interesting and useful operation of composition also requires that the indeterminant be stated.
Definition Let p and q be polynomials in x.
Suppose and . But some polynomials commute with each other. Find all the linear polynomials that commute with . Show that if p and q are linear polynomials which commute with , then p and q commute. Show that does not commute with any polynomial of degree higher than 1.