$x^6+2x^3-3x^2+1$, irreducible over $\mathbb{Q}$.
I am trying to determine whether or not the above polynomial is irreducible over the specified field, $\mathbb{Q}$.
Some tools I have are: Eisenstein's Criterion, reduction $\mod p$ (where p is a prime), the rational roots theorem, among other smaller tricks. Eisenstein's Criterion cannot be applied since the $\gcd(2,3)=1$ which divides the leading coefficient. Moreover, the rational roots theorem asserts only that the polynomial does not reduce into a polynomial of degree $1$ and one of degree $5$; still, the polynomial might reduce. Therefore, I tried reduction $\mod p$.
I figured that $x^6+2x^3-3x^2+1 \equiv x^6 +2x^3+1\mod 3$. Then, replace $x^3=y$ so that we have $y^2+2y+1 \mod 3$. Unfortunately, this does factor since $[2]$ is a zero and therefore we have learned nothing about the original polynomial.
What is a different, better method of attack?
Best Answer
Hint:
Set $y=x+1\iff x=y-1$ and apply Eisenstein's criterion to the polynomial $P(y-1)$.