# Extremely Divisible Polynomial

Consider all degree 4 monic polynomials $$f(x)$$ with complex coefficients which satisfy the condition that for all positive integers $$n$$, $$f(x)$$ divides $$f(x^n)$$.

Over all such polynomials, what is the smallest possible positive integer value of $$f(-3)$$?

(As a followup, can you describe all polynomials such that $$f(x) \mid f(x^n)$$ for all positive integers $$n$$?)

×