<s>[INST] Is $x^4 + 2x^2 - x + 1$ irreducible in $\mathbb Z_7[x]$? How would one do this? 
I know since it doesn't have roots it can only be divisible by irreducible polynomials of degree 2. 
How would I prove that it is or it isn't? 
There are a lot of polynomials with this conditions.
 [/INST] The product of all the monic, irreducible polynomials over $\mathbb{F}_7$ wih degree $\leq 2$ is given by $x^{49}-x$, so if we prove that $f(x)=(x^2+1)^2-x$ and $x^{48}-1$ are coprime over $\mathbb{F}_7[x]$ we are done. Let us compute $x^{48}-1\pmod{f(x)}$, for instance through the Brauer chain $x^4\to x^8\to x^{16}\to x^{32}\to x^{48}$.
$$ x^4 \equiv -2x^2+x-1\pmod{f(x)} $$
$$ x^8 \equiv 3x^3-3x^2+2x-3\pmod{f(x)} $$
$$ x^{16} \equiv x^3+3x^2+2x-1\pmod{f(x)} $$
$$ x^{32} \equiv -x^3+2x^2+x-3\pmod{f(x)} $$
$$ x^{48}-1 \equiv -x^3-2x^2+x+2 \pmod{f(x)}$$
give
$$\gcd(x^{48}-1,x^4+2x^2-x+1)=\gcd(x^3+2x^2-x-2,x^4+2x^2-x+1) $$
$$ = \gcd(x^3+2x^2-x-2,-x-3) $$
but $x=4$ is not a root of $x^3+2x^2-x-2$ since $7\nmid 90$ and we have finished.
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