"Exhaustive" search (81 cases) sez there are no solvable irreducible quintics with coeffs restricted to {-1,0,1}. After x^5-2 (modulo root negation and reciprocation), the smallest, in some sense, seems to be -1 - 2 x - x^2 + x^3 + x^4 + x^5, whose five solutions are -1/5 +w*u_00+w^2*u_01+w^3*u_10+w^4*u_11, where w = e^(2 pi i n/5), n an integer mod 5, and u_jk := j j k (-1) 1371 - 145 (-1) (sqrt 5) + 65 (-1) sqrt(470 (1 - ------)) sqrt 5 R(-----------------------------------------------------------) 12500 where R is the *real* fifth root of a real: R(#):=#/(#^4)^(1/5). If you use the default complex 5th root, the negative ones will need a correction factor of e^(4 i pi/5). Beware: This is the kind of 5th root they mean in the Wikipedia article<http://en.wikipedia.org/wiki/Quintic_function> and in the examples in the (glitchy) Intelligencer article<http://www.math.carleton.ca/%7Ewilliams/papers/pdf/244.pdf> . (InputForm in case Gmail murdered the above: (1371 - 145*(-1)^j*Sqrt[5] + 65*(-1)^k*Sqrt[470]*Sqrt[1 - (-1)^j/Sqrt[5]])/12500 ). The u_jk^5 are the (real) roots of the quartic 1 + 325793750 u + 4331113281250 u^2 - 41839599609375 u^3 + 95367431640625 u^4 (Recall that the quintic only had 2s and 1s.) So a convenient way to exhibit the solution to an irreducible quintic ax^5+bx^4+... is the constant -b/(5a) + one of these quartics. Recall that quartic solutions in general have quadratic surds inside cubic surds inside quartic surds, so for the quintic we appear doomed to suffer quadratic surds inside cubic surds inside ... inside quintic surds. But no! For some magic Galwological reason, these quartics seem always to solve down to (real!) quadratics in quadratics, at worst. Sometimes, as with -3 + 3 x + 3 x^2 + 3 x^3 + 3 x^4 + x^5, the quartic even factors: (59049 - 12543750 u + 9765625 u^2) (-7776 + 1050000 u + 9765625 u^2). But usually it doesn't even decompose, so your quartic solver needs to be smart. --rwg Is there a name for these nice quartics? They're not usually biquadratic. How about "nice"?-) PS, for some reason, Preview saturates my Mac CPU while trying to scroll the later portions of the Intelligencer article<http://www.math.carleton.ca/%7Ewilliams/papers/pdf/244.pdf>.