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Let us consider one variable polynomials with the senior coefficient equal to one. We shall say that two polynomials

P(x)

and

Q(x)

commute, if

P(Q(x))=Q(P(x))

(i.e. we obtain the same polynomial, having collected the similar terms).
a) For every a find all

Q

such that the

Q

degree is not greater than three, and

Q

commutes with

(x^2 - a)

.
b) Let

P

be a square polynomial, and

k

is a natural number. Prove that there is not more than one commuting with

P

k

-degree polynomial.
c) Find the

4

-degree and

8

-degree polynomials commuting with the given square polynomial

P

.
d)

R

and

Q

commute with the same square polynomial

P

. Prove that

Q

and

R

commute.
e) Prove that there exists a sequence

P_2, P_3, ... , P_n, ...

(

P_k

k

-degree polynomial), such that

P_2(x) = x^2 - 2

, and all the polynomials in this infinite sequence pairwise commute.

Problem Statement