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?

Thank you in advance!


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.

  • $\begingroup$ Thank you! Although we're aren't yet introduced in Galois theory, this solution sounds seems to be most reasonable. $\endgroup$ – nakajuice Dec 11 '13 at 23:38

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