I want to know why if $a$ is odd, then the polynomial $$ 3x^5 -4x^4+2x^3+x^2+18x+a $$ is irreducible in $\mathbb{Z}[x]$.

I know that if it factors into a polynomial of degree 4 and one of degree 1, it means that it has an integer root, but that is a contradiction since any integer lets an odd value, so it can't be even (0). How can I show that it can't be factorised into a polynomial of degree 2 and another of degree 3?


Hint 1

Consider your polynomial modulo $2$. You get a polynomial over the field $\{ 0, 1\}$ with two elements.

Hint 2

You will only have to check that $0, 1$ are not roots, and that it is not divisible by $x^2 + x + 1$.

  • $\begingroup$ How the second hint is useful? I see that, but I can't see how it solves the problem. $\endgroup$ – MonsieurGalois Oct 28 '16 at 19:29
  • $\begingroup$ You prove that the polynomial modulo $2$ is irreducible. hence the original polynomial is irreducible over the integers. $\endgroup$ – Andreas Caranti Oct 28 '16 at 19:36
  • $\begingroup$ But why is true that the polynomial must be irreducible in $\mathbb{Z}$ if it is irreducible modulo 2? $\endgroup$ – MonsieurGalois Oct 28 '16 at 19:47
  • 3
    $\begingroup$ Because if the polynomial is reducible in $\mathbb{Z}$... $\endgroup$ – Andreas Caranti Oct 28 '16 at 19:49

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