Question

Give an example of a polynomial in$\mathit{Z}\left(x\right)$ that is irreducible in$\mathit{Q}\left(x\right)$ but factors when reduced mod 2,3,4 and 5.

Short answer

The example is$180x^3+x^2-1$

Step-by-step solution

Write the example

Consider $180x^3+x^2-1$is irreducible in $Q\left[x\right]$as by rational root test, the root of the polynomial is as follows:

$\left\{\begin{array}{ll}\pm 1,\pm \frac{1}{2},\pm \frac{1}{3},\pm \frac{1}{4},\pm \frac{1}{5},\pm \frac{1}{6},\pm \frac{1}{9},\pm \frac{1}{10},\pm \frac{1}{12},\pm \frac{1}{15},\pm \frac{1}{18},& \\ \pm \frac{1}{20},\pm \frac{1}{30},\pm \frac{1}{36},\pm \frac{1}{45},\pm \frac{1}{60},\pm \frac{1}{90},\pm \frac{1}{180}\left.\begin{array}{r}\\ \end{array}\right\}& \end{array}\right.$

Modulo

Take the modulo as. p=2,3,4,5 as$180x^3+x^2-1=\left(x-1\right)\left(x+1\right)$

Therefore, the polynomial has factorswhen reduced mod 2,3,4, and 5.

Hence, the example is$180x^3+x^2-1$.