Question 18
A polynomial is monic if its leading coefficient is $1$. Two polynomials $p(x)$ and $q(x)$ commute if $p(q(x)) = q(p(x))$.
(1) Find all monic polynomials of degree $3$ or less which commute with $x^2 - k$.
(2) Given a monic polynomial $p(x)$, show that there is at most one monic polynomial of degree $n$ which commutes with $p(x)^2$.
(3) Find the polynomials described in (2) for $n = 4$ and $n = 8$.
(4) If $q(x)$ and $r(x)$ are monic polynomials which both commute with $p(x)^2$, show that $q(x)$ and $r(x)$ commute.
(5) Show that there is a sequence of polynomials $p_2(x), p_3(x), \ldots$ such that $p_2(x) = x^2 - 2$, $p_n(x)$ has degree $n$ and all polynomials in the sequence commute.
(1) Find all monic polynomials of degree $3$ or less which commute with $x^2 - k$.
(2) Given a monic polynomial $p(x)$, show that there is at most one monic polynomial of degree $n$ which commutes with $p(x)^2$.
(3) Find the polynomials described in (2) for $n = 4$ and $n = 8$.
(4) If $q(x)$ and $r(x)$ are monic polynomials which both commute with $p(x)^2$, show that $q(x)$ and $r(x)$ commute.
(5) Show that there is a sequence of polynomials $p_2(x), p_3(x), \ldots$ such that $p_2(x) = x^2 - 2$, $p_n(x)$ has degree $n$ and all polynomials in the sequence commute.