Example 3 Find all non-zero polynomials $f(x)$ that satisfy $f\left(x^{2}\right)=(f(x))^{2}$.

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Example 3 Find all non-zero polynomials $f(x)$ that satisfy $f\left(x^{2}\right)=(f(x))^{2}$.
Solution: Let $f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}$, where $a_{n} \neq 0$. If at least one of the coefficients $a_{n-1}, \cdots, a_{1}, a_{0}$ is not 0, without loss of generality, let $k(k<n)$ be the largest integer such that $a_{k} \neq 0$. Thus, $f\left(x^{2}\right)=a_{n} x^{2 n}+a_{k} x^{2 k}+\cdots+a_{1} x^{2}+a_{0}$ $\equiv\left(a_{n} x^{n}+a_{k} x^{k}+\cdots+a_{1} x+a_{0}\right)^{2}=(f(x))^{2}$. By comparing the coefficients of $x^{n+k}$ on both sides of the equation, we get $0=2 a_{n} a_{k}$, which contradicts $a_{n} \neq 0, a_{k} \neq 0$. Therefore, $a_{n-1}=\cdots=a_{1}=a_{0}=0$, and thus $f(x)=a_{n} x^{n}$. From the condition $a_{n} x^{2 n} \equiv f\left(x^{2}\right) \equiv (f(x))^{2} \equiv a_{n}^{2} x^{2 n}$, we get $a_{n}=1$.
Therefore, $f(x)=x^{n}, n \in Z^{+}$.