This post is rather long, but the intro is necessary, and the question is at the end. Here's a "natural" solvable 17-th deg eqn with small coefficients: x^17-6 x^16-24 x^15-42 x^14-31 x^13-23 x^12-7 x^11-x^10-4 x^9-11 x^8-7 x^7-13 x^6-x^5+x^3+x^2+x-1 = 0 (eq.1) Its unique real root is exactly given by (in Mathematica) as x = zeta_48 DedekindEta[tau]/(Sqrt[2]DedekindEta[2tau]) = 9.1630942... with the root of unity zeta_48 = exp(2Pi I/48), tau = (1+Sqrt[-d])/2, and d = 383. This d has class number h(-d) = 17. To solve this, depress eq.1 (get rid of its x^(n-1) term), by letting x = (y+6)/17 to get, y^17-11832 y^15-1124346 y^14-55393735 y^13-1784741617 y^12-41171464807 y^11-711423456455 y^10-9455898295636 y^9-99724287747103 y^8-887992943070295 y^7-7665207188897171 y^6-70479807472769473 y^5-592167373130143650 y^4-3496187093606980919 y^3-8695712981307573757 y^2+68265051092799270505 y-427806967360317821039 = 0 (eq.2) Its 16-deg resolvent, a polynomial with INTEGER coefficients, call this R_16, has roots, z_k = ((y1 + w^k y2 + w^(2k) y8 + w^(3k) y7 + w^(4k) y16 + w^(5k) y4 + w^(6k) y12 + w^(7k) y15 + w^(8k) y11 + w^(9k) y10 + w^(10k) y14 + w^(11k) y13 + w^(12k) y5 + w^(13k) y17 + w^(14k) y6 + w^(15k) y9 + w^(16k) y3)/17)^17 for k = {1 to 16} where w is any complex 17th root of unity, and y_n follows the root object Root[poly, n] ordering in Mathematica. Approximately, these are, y1 = 149.7726 {y2, y3} = -27.62 -/+ 18.49i {y4, y5} = -21.61 -/+ 7.52i {y6, y7} = -16.58 -/+ 6.34i {y8, y9} = -10.57 -/+ 15.32i {y10, y11} = -5.02 -/+ 13.71i {y12, y13} = -2.34 -/+ 13.15i {y14, y15} = 2.57 -/+ 2.60i {y16, y17} = 6.31 -/+ 7.04i R_16 has *extremely* large integer coefficients, with the largest being the 248-digit constant term 429534618434587^17 which, naturally enough, is a 17th power. (Note: R_16 can easily be formed using 500-digits precision or more on the y_n, and multiplying the 16 factors together to form the polynomial.) The polynomial R_16 can be factored into two octics over the ext Sqrt[17]. This, in turn, can be factored into 2 quartics over Sqrt[2(17+Sqrt[17])]. This can be factored further into 2 quadratics using an expression involved in the 17th root of unity. (Apparently, to solve R_16 = 0, only square roots of square roots of square roots, etc, are needed.) The real root of eq.2 in radicals is then, y1 = z_1^(1/17) + z_2^(1/17) + z_3^(1/17) + … + z_16^(1/17) = 149.7726… *Problem*: Express the roots of R_16 purely in terms of the complex 17th root of unity. (If anyone knows how to contact the mathematician Peter-Lawrence Montgomery, he probably will know how, since he has done something similar with a septic root and the 29th root of unity.) Sincerely, - Tito