I am trying the fallowing exercise : Solve $P(X^2 -2)=P(X)^2 -2$ with P a monic polynomial (non-constant)
My attempt :
Let P satisfying $P(X^2-2) = (P(X))^2-2$
Then $Q(X)=P(X^2-2) = (P(X))^2-2$
Therefore, $$Q(X^2-2) = (P(X^2-2))^2-2 = (P(X)^2-2)^2-2 = Q^2-2$$
As X is a solution, by defining the sequence:
$(P_n)_{n \geq 1}$ with $P_1 = X$ and for all $n \geq 1, P_{n+1} = P_n^2-2$
We obtain a sequence of polynomials which are solutions.
But I don't know how to prove it's the only one. If someone have an idea to prove it or an another method to solve the problem ?
Thank you in advance for your time.