# Show that $x^4 + 4x^3 - 2x^2 - 5$ is irreducible over the rational numbers.

I need to show that $$f(x) = x^4 + 4x^3 - 2x^2 - 5$$ is irreducible over the rational numbers.

So was trying to use the Eisenstein's criterion, but I can't find such prime number p (I think it doesn't exist). I could possibly show that $$f(ax + b)$$ is irreducible. Any tips?

• Is there a typo in the title and the question? Do you intend $...4x^3...$ rather than $...4^3...$? Jun 14, 2021 at 8:28
• fixed that! Tanks! Jun 14, 2021 at 8:30
• The rational root theorem can be used to determine if there are rational roots. However, if I understand your question correctly, that is inconclusive. That is, you could (still) have $f(x) = (x^2 + ax + b)(x^2 + cx + d)$ where $a,b,c,d$ are all rational, but $f(x)$ still doesn't have a rational root. Therefore, as I say, I think the rational root theorem is inconclusive here. Jun 14, 2021 at 8:31

The polynomial is already irreducible over $$\Bbb F_3$$, where it is $$f=x^4+x^3+x^2+1.$$ So it is also irreducible over $$\Bbb Z$$ and $$\Bbb Q$$.

How is this fourth degree polynomial irreducible over $$\Bbb F_3$$? Because it doesn't have roots in $$\Bbb F_3$$ and because it is not a product of two irreducible monic polynomials of degree $$2$$ over $$\Bbb F_3$$ as well. This is easy here, because we only have $$3$$ such candidates: $$x^2+1,x^2+x−1$$ and $$x^2−x−1$$.

• I am totally ignorant in this area. Briefly, what is $\Bbb{F_3}$? Jun 14, 2021 at 8:33
• It is the finite field with $3$ elements. Do you know the mod $p$ irreducibility test? It is a standard result like Eisenstein. It is also called reduction crtiterion. Jun 14, 2021 at 8:35
• Thanks for the response. Although I'm not familiar with the mod $p$ irreducibility test, I can certainly do research myself. I've actually never studied Abstract Algebra, believe it or not; I merely occasionally glanced at a few pages now and then. You have given me more than enough to go on. It's not really appropriate for you to attempt more; that would constitute your filling the gap. It's now on me to go thru Abstract Algebra pdf's, which are readily available, and then later post mathSE questions, (if any). Jun 14, 2021 at 8:38
• But what p did you use to get $x^4 + x^3 + x^2 +1$ ? Jun 14, 2021 at 8:40
• No problem, I just wanted to be helpful, so that you can checkmark my answer :) I used $p=3$, so $\Bbb F_3=\Bbb Z/3\Bbb Z$, that was your first question. Jun 14, 2021 at 8:40

I could possibly show that $$f(ax + b)$$ is irreducible. Any tips?

Yes that works, notice that $$f(x-1)=x^4-8x^2+12x-10$$ is Eisenstein with respect to $$p=2$$, hence $$f(x)$$ is irreducible over rationals.