Galois Group of $x^p-x+a$ over $\mathbb{F}_p$ for prime $p$ and $a\neq 0\in \mathbb{F}_p$

Question

Question is to find Galois group of $f(x)=x^p-x+a$ over $\mathbb{F}_p$ for prime $p$ and $a\neq 0\in \mathbb{F}_p$

What i have done so far is :

I could see that $f(x)$ is Irreducible and separable...

Now, I need to find splitting field of $f(x)$

Suppose $b$ is a root of $f(x)$ in some extension field $E$ of $\mathbb{F}_p$

i.e., $b^p-b+a=0$

Consider $f(b+1)=(b+1)^p-(b+1)+a=b^p+1-b-1+a=b^p-b+a=0$

So, for similar reasons $f(b+i)$ is root..

So, if i adjoin one root then it would automatically contain all roots..

So, splitting field of $f(x)$ would be $\mathbb{F}_p(b)$

Now there is some little gap..

I would like to say that as $\mathbb{F}_p(b)$ is just a finite extension we see $\mathbb{F}_p(b)=\mathbb{F}_{p^p}$

We know that Galois group of finite extension is cyclic ...

In this case we see that Galois group is generated by $\sigma$ defined as $\sigma(\alpha)=\alpha^p$

I just want to make sure there are no Gaps in this..

Please help me to make this perfect...