Show a polynomial is irreducible

Question

I'm working through the proof of Hasse's theorem and I think I need to show that the polynomial $x^4 - 2ax^2 - 8bx + a^2$ is irreducible over $\mathbb{F}_p$, where $a$, $b$ are integers and $p$ is prime. Wolfram Alpha repliably informs me that it is but I can't see how to show it?

Answer 1

This cannot be true as stated. Take $a=b=1$ and $p=2$. Then $$ x^4-2ax^2-8bx+a^2=x^4-2x^2-8x+1=x^4+1=(x+1)^4. $$ Also, if you don'tt like $p=2$, this is reducible for $p=5$: $$ x^4-2x^2-8x+1=(x^3 + 3x^2 + 2x + 3)(x + 2). $$

And, of course, for $a=b=0$ is is obvious, too.