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?

  • $\begingroup$ Is there a typo in the title and the question? Do you intend $...4x^3...$ rather than $...4^3...$? $\endgroup$ Jun 14, 2021 at 8:28
  • $\begingroup$ fixed that! Tanks! $\endgroup$ Jun 14, 2021 at 8:30
  • $\begingroup$ 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. $\endgroup$ Jun 14, 2021 at 8:31

2 Answers 2


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 $x^2−x−1$.

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    $\begingroup$ I am totally ignorant in this area. Briefly, what is $\Bbb{F_3}$? $\endgroup$ Jun 14, 2021 at 8:33
  • $\begingroup$ 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. $\endgroup$ Jun 14, 2021 at 8:35
  • $\begingroup$ 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). $\endgroup$ Jun 14, 2021 at 8:38
  • $\begingroup$ But what p did you use to get $x^4 + x^3 + x^2 +1$ ? $\endgroup$ Jun 14, 2021 at 8:40
  • $\begingroup$ 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. $\endgroup$ 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.


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