Irreducibility, Separability, and Galois group of polynomials over finite fields

Let $f$ be a polynomial of degree $n$ over the finite field $\mathbb{F}_p$. We aim to find its Galois group.

If $f$ is irreducible and separable, then $\mathbb{F}_{p^n}$ is its splitting field, and its Galois group is $Z/nZ$, which is generated by $\sigma:x\mapsto x^p$. Now, suppose we do not know about irreducibility and separability. One way to find out is to consider $\alpha$ a root of $f$ and find $\{\alpha,\sigma(\alpha),\cdots,\sigma^{n-1}(\alpha)\}$; if this set has $n$ distinct elements, then $f$ is irreducible and separable with the Galois group $Z/nZ$. If not, then it gives an idea of how we can factor $f$.

Now, my main problem is: if we have $\{\alpha,\sigma(\alpha),\cdots,\sigma^{n-1}(\alpha)\}$, how can we find out whether they are distinct or not?

For example, consider $f=x^6+x+1$ over $\mathbb{F}_2$. I do not know any easy way to find the irreducibility and separability of this polynomial except by applying the Frobenius map. Do you know any?

If $\alpha$ is a root of this polynomial, by applying the Frobenius map we get: $$\{\alpha,\alpha^2,\alpha^4,\alpha^2+\alpha^3,1+\alpha+\alpha^4,1+\alpha^3\}.$$ If I do not know a priori that this polynomial is irreducible and separable, am I able to say this set contains $6$ distinct elements?

By just looking at them, they seem to be distinct. However, I am afraid we may repeat a root just in a different form, for example $\alpha=\alpha^2+\alpha^3$, $\alpha^2=1+\alpha+\alpha^4$, and $\alpha^4=1+\alpha^3$. Is my concern valid? or this situation never happen?

If it happens, how can we handle it?