Let $p$ be a prime number , $f\in \mathbb Z[X]$ an irreducible polynomial with degree $p$ and coefficients in the range $[-1,1]$. Then the galois group of $f$ over $\mathbb Q$ is $S_p$
Can anyone prove or disprove this conjecture ?
The conjecture is true for the primes upto $p=11$. To prove the conjecture it would be sufficient to prove that the galois group of $f$ over $\mathbb Q$ contains a transposition, which is surely the case if there are exactly two non-real roots. But in general, I do not know how this can be shown.