Can I use Eisenstein's criterion to show $x^{4}-2x^{3}+2x^{2}+x+4$ is reducible over $\mathbb{Q}$?

Question

Can I use Eisenstein's criterion to show $x^{4}-2x^{3}+2x^{2}+x+4$ is reducible over $\mathbb{Q}$?

Can I say that there does not exist a prime that divides both $2$ and $1$? Or is there another way to show it is reducible? Do I just use rational root test?

Answer 1

You can check this yourself. The basic theorem is that, if this factors over the rationals, in fact it factors over the integers.

The second result states that if a non-constant polynomial with integer coefficients is irreducible over the integers, then it is also irreducible if it is considered as a polynomial over the rationals.

This is called Gauss's Lemma; one of them, anyway. Writing as $(x^2 + ax + c)(x^2 + bx + d),$ we find $cd = 4,$ so we have some four possibilities to investigate: $$ (x^2 + ax + 1)(x^2 + bx + 4),$$ $$ (x^2 + ax + 2)(x^2 + bx + 2),$$ $$ (x^2 + ax - 1)(x^2 + bx - 4),$$ $$ (x^2 + ax - 2)(x^2 + bx -2),$$ where, in each one, we see if we can find integer values for $a,b$ that make the multiplication come out as your $x^4 - 2 x^3 + 2 x^2 + x + 4. $ It is pretty easy (in each of four) to find the coefficient of $x^3$ as a combination of $a,b,$ it is also easy to find the coefficient of $x.$ If any of the four gives integer values, the final thing is to see how the coefficient of $x^2$ comes out.