Is it possible to apply a shift (to the variable $x$) and Eisenstein's criterion to show that the polynomial $f(x) = x^3 + x^2 − 2x − 1$ is irreducible over the rationals?
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Isn't it easier to just check for rational roots? – lulu Jan 29 '21 at 17:39
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@Surb thanks. can you explain? – Heitor Fontana Jan 29 '21 at 17:40
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@lulu yes, I agree but I want to know if I can use Eisenstein (with a shift) – Heitor Fontana Jan 29 '21 at 17:41
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You can shift by 9 to get $x^3 +28x^2+259x+791$ which is irreducible by looking at divisibility by 7.
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$9\equiv\text{something smaller}\bmod 7$. Why did you not try that smaller shift? (The smaller shift also gives a square-free constant term.) – Oscar Lanzi Feb 10 '21 at 21:33