From the 3rd edition of the book "The Linear Algebra a Beginning Graduate Student Ought to Know" by Jonathan S. Golan, we find the following exercise under chapter 4:
Exercise 140:
"Is the polynomial $X^7 + X^4 + 1 ∈ Q[X]$ irreducible?"
At this point in the book, we have been shown only the Eisenstein criterion and the "$X+c$" substitution trick in order to exploit the criterion. However, the Eisenstein criterion trick won't work as $1$ and $-1$ don't work as substitutions for $c$, and any greater choice of an integer for $c$ we shift the polynomial in such a way so as to yield a constant term of the form $mc + 1$ where $m ∈ Z$, hence we will fail the criterion.
So my question is, is there any elementary way to show that $X^7 + X^4 + 1$ is irreducible or to show that it is not?
Best Answer
Mod 2, there are 2 irreducible cubics and 1 irreducible quadratic, and none divide your polynomial by just checking. So the polynomial is irreducible mod 2, hence over rationals.