Let $p$ be a prime number, and let $f(x)$ be a polynomial with integer coefficients, whose leading coefficient is not divisible by $p$. Prove that the congruence

$$f(x) \equiv 0 \quad(\bmod p)$$

has at most $d$ solutions, where $d$ is the degree of $f(x)$.

Deduce that all coefficients of the polynomial

$$x^{p-1}-1-((x-1)(x-2) \cdots(x-p+1))$$

must be divisible by $p$, and prove that:

(i) $(p-1) !+1 \equiv 0 \quad(\bmod p)$;

(ii) if $p$ is odd, the numerator of the fraction

$$u_{p}=1+\frac{1}{2}+\cdots+\frac{1}{p-1}$$

is divisible by $p$.

Assume now that $p \geqslant 5$. Show by example that (i) cannot be strengthened to $(p-1) !+1 \equiv 0 \quad\left(\bmod p^{2}\right)$.