TITLE: Roots of $x^2 + 2x + 2$
QUESTION [5 upvotes]: I'm trying to show that there are infinitely many values of $p$ such that $x^2 + 2x + 2$ has no roots over $\mathbb{F}_p$.  Is this easily solvable?  (I kind of came up with it myself so I don't know.)
Thanks!

REPLY [3 votes]: I would give a slightly different group theoretic perspective. If $F$ is the field of $p$ elements and $p$ is an odd prime, then the multiplicative group of non-zero elements of $F$ has order $p-1.$ If there is an element $a \in F$ such that $a^2 = -1,$ then $a^4 =1$ and the cyclic group $\langle a \rangle$ has order $4.$ By Lagrange's theorem, $4$ divides $p-1,$ so $p$ has the form $4k+1$ for some integer $k.$