I am attempting to solve part (d) of the following problem from Artin's algebra book. I have already solved parts (a) through (c).  (a) Let $p$ be an odd prime. Prove that exactly half of the elements of $\mathbb{F}_p^\times$ are squares and that if $α$ and $β$ are nonsquares, then $αβ$ is a square. (b) Prove the same assertion for any finite field of odd order. (c) Prove that in a finite field of even order, every element is a square. (d) Prove that the irreducible polynomial for $γ=\sqrt2+\sqrt3$ over $\mathbb{Q}$ is redubile modulo $p$ for every prime $p$.  Now, I have shown that $g(x)=x^4-10x^2+1\in \mathbb{Q}[x]$ is the irreducible polynomial for $γ$. Let $p$ be a prime. Let $f(x)=x^4-10x^2+1 \in \mathbb{F}_p[x]$. I want to show $f$ is reducible. I see that either 2, 4, or 6 is equal to $s^2$ for some $s\in\mathbb{F}_p$. I think it will be useful to proceed by cases here. It someone could show me a complete, elementary, simple proof for the $s^2=2$ case, I should be able to finish the rest on my own.
So I have seen the similar question and answers on here for $x^4 +1$, but I am having trouble extending anything there to this polynomial... I understand it is fairly trivial with Galois theory, but my class has just barely covered Field Extensions, so suffice it to say we have no Galois theory to play with.  I managed to prove it for the primes such that $p \equiv 1, 7 \pmod 8$, by noting that $2$ is a square modulo those primes and thus $x^4 - 2x^2 +1 = (x^2 -1 + 2qx)(x^2 - 1 - 2qx)$ for those $\mathbb{Z}_p$... however, trying to get a similar result for $3 \pmod 8$ and $5 \pmod 8$ has been stumping me for a long time, I am having a hard time making $q^2 = -1$ and $q^2 = -2$ give me something factorable...  I guess the worst part of all of this is that I don't think this solution is even particularly enlightening, in terms of abstract algebra. It's really just some number theory trickery. I don't think my course has prepared me theoretically for this problem, does anyone have an elementary approach to it?
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