$x^4+x+1 \in\mathbb{Z}_2[x]$ - Galois group

Question

I do not understand very well about extensions of fields $\mathbb{Z}_p$. I understood about field's extensions of $\mathbb{Q}$, for instance, $\mathbb{R},\mathbb{C}$ etc.

But, what fields extend $\mathbb{Z}_p$?

Because of these uncertaintes, I cannot solve the following problem:

Determine the Galois group of the extension $\Sigma:\mathbb{Z}_2$, where $\Sigma$ is the splitting field of $f(x)=x^4+x+1\in\mathbb{Z}_2[x]$ over $\mathbb{Z}_2$.

Answer 1

Check out the following document: http://www.math.uconn.edu/~kconrad/blurbs/galoistheory/finitefields.pdf