Is $x^4+2x^2-8m^2x+1$ irreducible over $\mathbb{Q}$ for $m \in \mathbb{Z}$, $m>0$?
Wolfram alpha says it is but gives no proof why. Eisenstein's criterion doesn't apply. I have no idea how to proceed. Can anyone help?
2026-04-05 20:52:34.1775422354
Is $x^4+2x^2-8m^2x+1$ irreducible over $\mathbb{Q}$?
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Clearly there are no integer (hence no rational roots), so we need to only check for quadratic factors. So we need to only check for the form $$x^4+2x^2-8m^2x+1=(x^2+ax\pm1)(x^2-ax\pm1)$$ which gives $2-a^2=2$ and $\pm2a=-8m^2$ which is possible only when $a=m=0$.