Eisenstein's Criterion with an example

Question

Wolfram Alpha says $x^5 -x^2 +1$ is irreducible over $\mathbb{Z}$. Is there any way to prove it by Eisenstein's Criterion?

I tried to translate this function. I translated the function a couple of times by some integers, but none of the translations worked. Any hint would be appreciated. Thanks so much.

Answer 1

By here an Eisenstein shift cannot work. But it follows easily from irreducibility mod $2\!:\,$ over $\Bbb F_2$ it has no roots so no linear factors, so if it splits it has an irreducible quadratic factor $g$, therefore in $\, \Bbb F_2[x]/g \cong \Bbb F_{\color{#c00}4}\!:\,$ $\,\color{#c00}{x^3 = 1}\,$ so $\ 0 = f = x^2(\color{#c00}{x^3})-x^2+1 = 1,\,$ contradiction.

Remark $ $ Above is a special case of a general polynomial irreducibility test over finite fields - which is an an efficient analog of the impractical Pocklington-Lehmer integer primality test.

Answer 2

Hint: Expand $ (x+a)^5 - (x+a)^2 + 1$

Compare the coefficients of $x^4 , x^2$.

Show that $ \gcd ( 5a, 10a^3 - 1) = 1$.

$ $

Hence conclude that Einstein's criterion cannot prove it is irreducible.