RU-ALLRUSSIAN 1977 Problem 17

A polynomial is monic if its leading coefficient is 1. Two polynomials $\mathrm{p(x)}$ and $\mathrm{q(x)}$ commute if $\mathrm{p(q(x)) = q(p(x))}$ .

(1) Find all monic polynomials of degree 3 or less which commute with $\mathrm{x}^2 - \mathrm{k}$ .

(2) Given a monic polynomial $\mathrm{p(x)}$ , show that there is at most one monic polynomial of degree n which commutes with $\mathrm{p(x)^2}$ .

(3) Find the polynomials described in (2) for $\mathrm{n} = 4$ and $\mathrm{n} = 8$ .

(4) If $\mathrm{q(x)}$ and $\mathrm{r(x)}$ are monic polynomials which both commute with $\mathrm{p(x)^2}$ , show that $\mathrm{q(x)}$ and $\mathrm{r(x)}$ commute.

(5) Show that there is a sequence of polynomials $\mathrm{p}_2(\mathrm{x}), \mathrm{p}_3(\mathrm{x}), \ldots$ such that $\mathrm{p}_2(\mathrm{x}) = \mathrm{x}^2 - 2$ , $\mathrm{p}_n(\mathrm{x})$ has degree n and all polynomials in the sequence commute.