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I know that $p$ is separable when the discriminant is not zero, $p'$ and $p$ don't have a common root, $p$ is irreducible, but that's it. Is there a criterion over $\mathbb Q$ for separability in terms of the coefficients similar to Eisenstein or related criteria for irreducibility?

I am asking this since my conjecture is that polynomials $1+\sum_{a} t^a$ seem to be separable where $a$ runs through a nonempty set of distinct odd numbers.

Thanks for your help.

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  • $\begingroup$ Whats ur ground field $\endgroup$ – Ignorant Mathematician Apr 1 at 6:20
  • $\begingroup$ over the rationals $\endgroup$ – orgesleka Apr 1 at 6:21

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