Field extensions and irreducible polynomials

Question

Let $F=Z_p$ and $f = x^p - x -b$ be a irreducable polynomial under $F$. If E is the splinting field of $f$ and root $u \in E$ how can we show that $E = F(u)$.

Answer 1

All the roots of $f$ are $u, u+1, u+2,....,u+p-1$ each of which $\in F(u)$. Hence the conclusion.

Observe that if $i\in \mathbb{Z}_p$ then $(u+i)^p-(u+i)-b=u^p+i^p-u-i-b=u^p+i-u-i-b=u^p-u-b=0$ since $u$ is a root.