Why is $f(x) = x^3+x+3$ irreducible over $\mathbb{Q}$?

Question

I could probably figure this out using analysis, but all the algebraic irreducibility tests that I know of - pretty much just Eisenstein's Criterion and the equivalence of irreducibility over $\mathbb{Q}$ and $\mathbb{Z}$ - cannot be applied to this $f$.

Any help towards an algebraic plan of attack would be appreciated!

Answer 1

You only have to show that the polynomial has no integer root because the polynomial has leading coefficient $1$.

You only need to verify the divisors of $3$, namely $-3,-1,1,3$.

Answer 2

The degree of $f$ is $3$. Then, if $f$ were not irreducible, it would have a divisor of degree one, and hence, a rational root.