(=0) solvable in radicals? SPOILER: Sure. It's cyclotomic, so x=exp(2 pi k/7) . But wait--what are 7th roots (of unity, or anything) doing in the solution of a sextic? Especially a reciprocal one. The roots should be (bi?)cubic, at worst. Indeed, here's all six. -1/6 + 1/6 E^(I (1/6 + #1) \[Pi]) Sqrt[7] E^(-((2 I \[Pi])/ 3)) + -(((E^((I 5 \[Pi])/6) (7/3)^(1/6) (9 I - 13 Sqrt[3])^( 1/3)) E^(-((I \[Pi])/3)) (E^((I \[Pi])/3))^#1)/( 6 2^(1/3))) + ( 7^(1/3) (E^((I \[Pi])/6) (-I - 3 Sqrt[3])^(1/3)) E^((2 I \[Pi])/ 3) (E^(-((2 I \[Pi])/3)))^#1)/( 6 2^(1/3)) + (((-(7/3))^(1/6) (-9 I - 13 Sqrt[3])^(1/3)) E^(-(( 2 I \[Pi])/3)) (-E^(((2 I \[Pi])/3)))^#1)/(6 2^(1/3)) + 1/6 7^(1/3) (-E^(((I 5 \[Pi])/6)) (1/2 (I - 3 Sqrt[3]))^(1/3)) E^(-(( 2 I \[Pi])/3)) (E^((2 I \[Pi])/3))^#1 & /@ Range[6]; Check: Rationalize[Abs[#]]*E^(\[Pi]*I*Rationalize[Arg[#]/\[Pi]]) & /@ N[%, 33] {E^((2 I \[Pi])/7), E^((6 I \[Pi])/7), E^(( 4 I \[Pi])/7), E^(-((2 I \[Pi])/7)), E^(-((6 I \[Pi])/7)), E^(-((4 I \[Pi])/7))} So we can express E^((2 I \[Pi])/7) in terms of E^((I \[Pi])/3) and cubic surds. http://mathworld.wolfram.com/TrigonometryAngles.html gives orders and, in prime cases, explicit formulas for the minimal polynomials for sin|cos(2 pi/n). Is it obvious why these (degree < n) polynomials always solve in radicals? (I.e., how does Developer`TrigToRadicals work?) Also, is there a solvable sextic trinomial which is neither bicubic nor triquadratic? An all-night (very inefficient) brute force search is so far fruitless. An even stronger conjecture: There are no true 6th roots of surds in the solutions of sextics! I.e., they always "denest" down to cuberoots. E.g., from that Cos[Pi/21] I sent yesterday, DenestRadicals2[(1/2 I (13 I + 3 Sqrt[3]))^(1/6)] -(-1)^(5/6) (1/2 (I - 3 Sqrt[3]))^(1/3) Even in the monster resolvent 2925951033851274156588135512485165232256823853056 - 2697817290737324449800236848640264992467435520 #1 + 9932351343021963689693473396732415411860992 #1^2 - 1881654619801628210127689611068977299937 #1^3 + 11450425009897563891465337536606118710 #1^4 + 3847649781964086608961673413540069 #1^5 + 378818692265664781682717625943 #1^6 of 8903 + 47647 v + 39672 v^2 + 7192 v^3 - 522 v^4 - 174 v^5 + v^7, the Montgomery|Piezas septic: DenestRadicals2[(1/2 (83845278844261475531122300164572695153 - 19718776166730709919674319096710083663 I Sqrt[3]))^(1/6)] ((-(1/3))^(1/6) (-22872248663753176389 I - 2586380087496171401 Sqrt[3])^(1/3))/2^(1/3) --rwg