Take, for example, $ X^4 + 2X + 2 $ in $ \mathbb{Q}[X] $. How do I determine if this is irreducible?
Thoughts:
I know Gauss' Lemma and Eisenstein's criterion, but they only work for primitive polynomials.
Take, for example, $ X^4 + 2X + 2 $ in $ \mathbb{Q}[X] $. How do I determine if this is irreducible?
Thoughts:
I know Gauss' Lemma and Eisenstein's criterion, but they only work for primitive polynomials.
Eisenstein's criterion works for proving that non-primitive polynomials are irreducible in $\mathbb{Q}[x]$. One then might hope to use Gauss's Lemma to prove that a polynomial with integer coefficients which is irreducible in $\mathbb{Q}[x]$ is also irreducible in $\mathbb{Z}[x]$, which then requires primitivity. So, because
by Eisenstein's criterion the polynomial $x^4+2x+2$ is irreducible in $\mathbb{Q}[x]$.
In fact, the polynomial $x^4+2x+2$ is primitive, since $\gcd(1,0,0,2,2)=1$, so the polynomial is irreducible in $\mathbb{Z}[x]$ too.