Irreducibility criterions for polynomials over $\mathbb{Z}$ or $\mathbb{Q}$

Question

Given a polynomial with "large" coefficients and powers over $\mathbb{Z}$ or $\mathbb{Q}$, how can we check the irreducibility of it?

For example, let us have the following polynomial in $\mathbb{Z}[X]$:

3072*x^42 + 65536*x^41 + 21504*x^40 + 107520*x^38 + 2293760*x^37 + 752640*x^36 + 1693440*x^34 + 36126720*x^33 + 11854080*x^32 + 15805440*x^30 + 337182720*x^29 + 110638080*x^28 + 96808320*x^26 + 2065244160*x^25 + 677658240*x^24 + 406594944*x^22 + 8674025472*x^21 + 2846164608*x^20 + 1185901920*x^18 + 25299240960*x^17 + 8301313440*x^16 + 2371803840*x^14 + 50598481920*x^13 + 16602626880*x^12 + 3112992540*x^10 + 66410507520*x^9 + 21790947780*x^8 + 2421216420*x^6 + 51652616960*x^5 + 16948514940*x^4 + 847425747*x^2 + 18078415936*x + 5931980230

In a second, sagemath says that it is irreducible. By hand, it is clearly not that easy.

Answer 1

Mathematica gives this simplification:

$$x \left(4 (x+21) (3 x+1) \left(x^4+7\right) \left(4 \left(4 x^{12}+70 x^8+490 x^4+1715\right) x^4+12005\right) \left(4 \left(2 x^4+7\right) \left(2 \left(x^4+7\right) x^4+49\right) x^4+2401\right) x^3+847425747 x+18078415936\right)+5931980230$$

Answer 2

There is the Eisenstein's criteria : let $P=a_0+\ldots+a_nX^n\in\mathbb{Z}[X]$ and $p$ a prime number such that : $$ \forall i\in[\![0,n-1]\!],\,p|a_i $$ $$ p\!\not| a_n $$ $$ p^2\!\not|a_0 $$ Then $P$ is irreductible in $\mathbb{Q}[X]$. Moreover, if $a_0\wedge\ldots\wedge a_n=1$, then $P$ is irreductible in $\mathbb{Z}[X]$ according to Gauss' lemma.

Proof : Let $$ \overline{P}=\overline{a_0}+\ldots+\overline{a_n}X^n\in\mathbb{Z}/p\mathbb{Z}[X] $$ Since $p|a_i$ for $i\in[\![0,n-1]\!]$, $\overline{P}=\overline{a_n}X^n$. Let us suppose there exists $Q,R\in\mathbb{Q}[X]$ such that $P=QR$, $\deg Q\geqslant 1$ and $\deg R\geqslant 1$. According to Gauss' lemma, we can suppose that $P,Q\in\mathbb{Z}[X]$. We samely define $\overline{Q}$ and $\overline{R}$ so that $\overline{P}=\overline{Q}\cdot\overline{R}$. Thus $\overline{Q}=\alpha X^k$ and $\overline{R}=\beta X^{n-k}$ for $(\alpha,\beta)\in(\mathbb{Z}/p\mathbb{Z})^2$ and $k\in[\![1,n-1]\!]$. Since $\overline{Q(0)}=\overline{R(0)}=0$ and $a_0=P(0)R(0)$, $p^2|a_0$ which is not. Finally $P$ is irreductible in $\mathbb{Q}[X]$.