Find all polynomials $P$ with real coefficients such that $P\left(X^{2}\right)=P(X) P(X-1)$.
Let $\lambda$ be a complex root of $P$. Then $\lambda^{2}$ is a root, so $|\lambda|=1$. Moreover, $(\lambda+1)^{2}$ is a root, so $|\lambda+1|=1$. Thus, $\lambda=j=e^{2 i \pi / 3}$, or $\lambda=\bar{j}$. The polynomial sought is therefore of the form $P=((X-j)(X-\bar{j}))^{n}=\left(1+X+X^{2}\right)^{n}$. Conversely, these polynomials work.