# Short method to prove the irreducibility of $x^7+21x^5+35x^2+34x-8$ over $\Bbb Q$

I am given a task to prove that polynomial $f=x^7+21x^5+35x^2+34x-8$ is irreducible over $\Bbb Q$.

In my algebra course we learnt reduction and Eisenstein criterion. Eisenstein doesn't seem to work out here, as well as reduction criterion in $\Bbb F_2$. So I could use reduction criterion with $\Bbb F_3$, but then I would have to prove irreducibility by around 20 polynomial divisions (divide $f$ by all irreducible polynomials of degree less than 4). It isn't that hard, but I wanted to find out whether there is a smarter method to irreducibility in this case?

Modulo $7$ you get the polynomial $x^7 - x - 1$. This is an Artin-Schreier polynomial, so it's irreducible. There are some proofs of irreducibility here.