a) If $F$ is a splitting field of the polynomial $x^4 + 1$ over $\mathbb{Z}_7$
b) If $F$ is the minimal extension of $\mathbb{Z}_3$ in which every polynomial $f \in \mathbb{Z}_3[x]$ of degress 2 has a root.
Everything I see in my notes involves a polynomial of the form $x^{p^n} -x$. There is nothing involving $x^{p^n} -1$ so I am a little lost on part a. I know that $F$ is an extension of $\mathbb{Z}_7$ where $x^4 + 1$ decomposes into linear factors such that $x^4 + 1 = c \prod _{i=1}^{4} ( X - a_i)$ where $c \in \mathbb{Z}_7$ and for each $i, (X-a_i) \in F$. EDIT: I actually see now that a field $\mathbb{F}_q, q=p^n$ is always a simple extension of $\mathbb{Z}_p$ by a root of some irreducible polynomial over $\mathbb{Z}_p$ so I guess my question would be is $x^4 + 1$ irreducible over $\mathbb{Z}_7$?
For part b, I want consider all polynomials of degree 2 with coeffients of 0,1,2 that have a root, i.e are reducible. I've tried to go about this by finding them and there were quite a few so I'm guessing finding them isn't necesary. I think this has someting to do with quadratic non-residues, but I could be wrong.