6・151 Let $k$ be a positive integer, find all polynomials
$$
p(x)=a_{0}+a_{1} x+\cdots \cdot+a_{n} x^{n},
$$

such that $a_{i}$ are real numbers $(i=0,1,2, \cdots, n)$, and satisfy the equation
$$
p(p(x))=[p(x)]^{k} \text {. }
$$
[Solution] If $p(x)=c$ (a constant), then
$$
c=c^{k} \text {. }
$$

Therefore, when $k=1$, $p(x)$ can be any constant;
when $k \neq 1$, $p(x)=0$ or $p(x)=1$.
If $p(x) \neq c$ (a constant), then $p(x)$ can take infinitely many values. Thus, there are infinitely many values $y$ such that
$$
p(y)=y^{k} .
$$

Hence, for all $x$, we have
$$
p(x)=x^{k} \quad(k \geqslant 1) .
$$