Example 1 Try to solve the congruence equation
$$x^{2} \equiv 73(\bmod 127)$$
First, it is easy to see that $p=127$ is a prime number. Moreover, we have $p \equiv 3(\bmod 4)$, $73 \equiv 1(\bmod 8)$. By the properties of the Legendre symbol and the Jacobi symbol, we have
$$\begin{aligned}
\left(\frac{73}{127}\right)= & \left(\frac{127}{73}\right)=\left(\frac{54}{73}\right)=\left(\frac{2}{73}\right)\left(\frac{3}{73}\right)^{3}=\left(\frac{3}{73}\right) \\
& =\left(\frac{73}{3}\right)=\left(\frac{1}{3}\right)=1
\end{aligned}$$

Therefore, equation (3) has a solution. From the previous discussion about case $I$, we immediately obtain the solution to the original congruence equation as
$$\begin{array}{l}
x_{0} \equiv \pm 73^{\frac{127+1}{4}}= \pm 73^{32} \equiv \pm(5329)^{16} \equiv \pm(-5)^{16} \equiv \\
\equiv \pm(390625)^{2} \equiv \pm(-27)^{2} \equiv \pm 729 \equiv \mp 33(\bmod 127)
\end{array}$$