TITLE: How to find all the roots in this ring?
QUESTION [2 upvotes]: Let $F:= \mathbb{F}_7[x]/(x^2+3x+1)$

Is it a field?
Find all the roots in F of the polynom  $f (Y) := Y^2+[3]_{F}Y +[1]_{F} \in F[Y]$.

Attempt:

It is a field, because $x^2+3x+1$ is irreducible $\in \mathbb{F}_7[x]$. In fact it has no roots $\in \mathbb{F}_7$.
I suppose I can't just replace numbers from $0$ to $6$ in the place of the $Y$. What should you do to solve this problem?

REPLY [3 votes]: Well, $f$ has $[x]$ as a root, the other root has to be $-3-[x]$ (Vieta).
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