# special polynomial

Is there a polynomial with coefficients -1,0,1 which is irreducible over the rationals and has the alternating group as its galois group ?

More concrete : Is there a polynomial of degree n, all coefficients -1,0 or 1, irreducible over the rationals with galois group A(n) ?

For small degrees, I enumerated the polynomials and none had the property. So I am curious if there are really none.

• It may take a while to fond one. Think of the almost, but not totally unrelated fact the the cyclotomic polynomials $\Phi_n$ have coefficients in $\{-1,0,1\}$ until you check $n=105$. – Hagen von Eitzen May 29 '13 at 21:39
• There might be some clues in math.stackexchange.com/questions/286944/… – Mark Bennet May 29 '13 at 21:44
• I checked the polynomials up to degree 8 and there is none. – user81078 Jun 5 '13 at 14:46