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What sufficient criteria are there for a polynomial to be separable?

1

$begingroup$

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.

polynomials separable-extension

share|cite|improve this question

edited Apr 1 at 9:31

stackExchangeUser

asked Apr 1 at 6:14

stackExchangeUserstackExchangeUser

1,236513

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Whats ur ground field
  $endgroup$
  – Ignorant Mathematician
  Apr 1 at 6:20
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  over the rationals
  $endgroup$
  – stackExchangeUser
  Apr 1 at 6:21
  

  
  
  
  

add a comment | 

1

$begingroup$

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.

polynomials separable-extension

share|cite|improve this question

edited Apr 1 at 9:31

stackExchangeUser

asked Apr 1 at 6:14

stackExchangeUserstackExchangeUser

1,236513

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Whats ur ground field
  $endgroup$
  – Ignorant Mathematician
  Apr 1 at 6:20
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  over the rationals
  $endgroup$
  – stackExchangeUser
  Apr 1 at 6:21
  

  
  
  
  

add a comment | 

$begingroup$

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.

polynomials separable-extension

share|cite|improve this question

edited Apr 1 at 9:31

stackExchangeUser

asked Apr 1 at 6:14

stackExchangeUserstackExchangeUser

1,236513

$endgroup$

share|cite|improve this question

edited Apr 1 at 9:31

stackExchangeUser

asked Apr 1 at 6:14

stackExchangeUserstackExchangeUser

1,236513

share|cite|improve this question

edited Apr 1 at 9:31

stackExchangeUser

asked Apr 1 at 6:14

stackExchangeUserstackExchangeUser

1,236513

asked Apr 1 at 6:14

stackExchangeUserstackExchangeUser

1,236513

asked Apr 1 at 6:14

stackExchangeUserstackExchangeUser

1,236513