# How to find all the roots in this ring?

Let $F:= \mathbb{F}_7[x]/(x^2+3x+1)$

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

Attempt:

1. It is a field, because $x^2+3x+1$ is irreducible $\in \mathbb{F}_7[x]$. In fact it has no roots $\in \mathbb{F}_7$.
2. 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?
• Since $Y$ is an element of the field $F$, $Y$ is a polynomial of degree at most one $1$. You want to find $Y=ax+b$ s.t. $f(Y)\equiv 0$. – Ragnar Jan 25 '14 at 15:42

Well, $f$ has $[x]$ as a root, the other root has to be $-3-[x]$ (Vieta).
• Could you please explayin why those are the roots? If I use the quadratic formula $D=b^2-4ac$ the result is $5$, then I need to calculate $\sqrt{5}$ and I can't find an element $x \in Z_{7}$ such that $x^2=5$ – Angelo Tricarico Jan 25 '14 at 17:48
• As I've said, use Vieta. By the way (but this is really irrelevant here), $\sqrt{5} \notin \mathbb{F}_7$ (but $\sqrt{5} \in \mathbb{F}_{7^2}$). – Martin Brandenburg Jan 25 '14 at 18:41
• The sum of the roots is $-3$ by Vieta. – Martin Brandenburg Jan 29 '14 at 8:51