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I am interested in an algorithm (or irreducibility criterion) for testing whether a cubic polynomial $f \in \mathbb{Q}[x]$ is reducible or not. By Gauss's Lemma, we can work with cubics in $\mathbb{Z}[x]$. Let $f = ax^3+bx^2+cx+d$, where $a,b,c,d \in \mathbb{Z}$. Suppose $M = \mathrm{max}\{a,b,c,d\}$. I want the run time of the algorithm to be $\mathrm{poly}(\log M)$. This rules out the rational root test, which takes around $O(M)$ time.

There is a general algorithm to factor integral polynomials called LLL algorithm which indeed runs in $\mathrm{poly}(\log M)$ time. However, I do not wish to invoke this heavy machinery and factor any degree polynomial. Is there a direct and efficient irreducibility testing algorithm, specifically for cubics, that is known in literature? By efficient, I mean $\mathrm{poly}(\log M)$-time algorithm.

One approach I can think of is to reduce this problem to testing whether $f$ (mod $p$) is reducible for a suitable prime $p$ and for finite fields, we indeed have Rabin's efficient reducibility testing algorithm. This reduction will clearly fail for degrees greater than 3, as we have examples of degree 4 irreducible polynomials in $\mathbb{Z}[x]$ which are reducible (mod $p$) for every prime $p$ (see this post). As per the post, for any degree 3 irreducible polynomial $f \in \mathbb{Z}[x]$, there exists a prime $p$ such that $f$ (mod $p$) remains irreducible, but I don't know the bound on $p$. Can we show $p \leq \mathrm{poly}(M)$? That will suffice for a $\mathrm{poly}(\log M)$-time algorithm.

Pranav Bisht
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  • An irreducible cubic has Galois group either $A_3$ or $S_3$, so at least one third of the elements are 3-cycles. I guess it may be possible to turn the effective version of Chebotarev/Frobenius density theorem into an estimate as to how high you need to check, but I'm prepared to be wrong about this. Of course, by carefully selecting $a,b,c,d$ you can make the modulo $p$ test fail for any finite set of primes. – Jyrki Lahtonen Jul 07 '21 at 11:20
  • @JyrkiLahtonen Thank you for your comment. Unfortunately, I have very little knowledge about Galois Theory. I have started to read it. Could you point me to some references/books for beginners that would be especially helpful with respect to this question and the density theorem you pointed out. – Pranav Bisht Jul 07 '21 at 13:14
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    What that suggests to me is something like the following. Pick a smallish prime $p$. Factor $f$ for irreducibility modulo $p$. If it is irreducible, then you are done, $f$ is irreducible over $\Bbb{Z}$ as well. If it has a repeated factor (here, a repeated root, something you can test with Euclid), then discard $p$ (when this happens $p$ is a factor of the discriminant and Frobenius/Chebotarev doesn't apply). Otherwise $p$ becomes a "witness" suggesting that $f$ may be reducible. The point is that if $f$ is irreducible, at most two thirds of the primes are witnesses. – Jyrki Lahtonen Jul 07 '21 at 14:10
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    (cont'd) So if you check $N$ witnesses without exiting the loop with definitive information about irreducibility, then the probability of reducibility can be estimated to be $1-(2/3)^N$. The effective bound should give something non-probabilistic. – Jyrki Lahtonen Jul 07 '21 at 14:14

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