Our sextic solver fervently assumes (I think with Piezas's blessing) that the solvable ones all factor into either cubics with quadratic surd coefficients, or vice versa. In[343]:= Factor[27*x^6 + 3087*x - 343, Extension -> Sqrt[21]] Out[343]= -(1/12) (-147 - 35 Sqrt[21] + (-21 + 21 Sqrt[21]) x + 6 Sqrt[21] x^2 - 18 x^3) * (147 - 35 Sqrt[21] + (21 + 21 Sqrt[21]) x + 6 Sqrt[21] x^2 + 18 x^3) is the former. The interesting real root is Out[218]= -Sqrt[(7/3)]/3 - 1/3 Sqrt[7/3] (1/2 (-70 + 9 Sqrt[21] + 9 Sqrt[85 - 12 Sqrt[21]]))^(1/3) + (Sqrt[7] (1 + 3 Sqrt[21]))/(3 2^(2/3) Sqrt[3] (-70 + 9 Sqrt[21] + 9 Sqrt[85 - 12 Sqrt[21]])^(1/3)) In[219]:= Factor[27*x^6 + 3087*x - 343 - 3^-9] Out[219]= ((-1 + 9 x) (6751270 + 9 x + 81 x^2 + 729 x^3 + 6561 x^4 + 59049 x^5))/19683 In[220]:= ContinuedFraction[%%, 33] Out[220]= {0, 9, 750141, 1, 1, 4, 50009, 4, 3, 4, 6, 1, 389, 1, 5, 7, 7, 2, 5, 1, 1, 1, 4, 64, 2, 1, 1, 1, 1, 1, 1, 2, 7} NDE>Are you suggesting there's some significance to the large convergents? Well, it can't hold a candle to Brillhart's In[228]:= ContinuedFraction[(15 Sqrt[3]-Sqrt[163])^(1/3)/Sqrt[3]+(15 Sqrt[3]+Sqrt[163])^(1/3)/Sqrt[3],239] Out[228]= {3,3,7,4,2,30,1,8,3,1,1,1,9,2,2,1,3,22986,2,1,32,8,2,1,8,55,1,5,2,28,1,5,1,1501790,1,2,1,7,6,1,1,5,2,1,6,2,2,1,2,1,1,3,1,3,1,2,4,3,1,35657,1,17,2,15,1,1,2,1,1,5,3,2,1,1,7,2,1,7,1,3,25,49405,1,1,3,1,1,4,1,2,15,1,2,83,1,162,2,1,1,1,2,2,1,53460,1,6,4,3,4,13,5,15,6,1,4,1,4,1,1,2,1,16467250,1,3,1,7,2,6,1,95,20,1,2,1,6,1,1,8,1,48120,1,2,17,2,1,2,1,4,2,3,1,2,23,3,2,1,1,1,2,1,27,325927,1,60,1,87,1,2,1,5,1,1,1,2,2,2,2,2,17,4,9,9,1,7,11,1,2,9,1,14,4,6,1,22,11,1,1,1,1,4,1,3,2,1,2,1,1,2,4,2,1,5,1,8,2,2,5,1,2,1,1,1,1,1,3,1,2,6,10,1,3,1,3,2,2,1,1,2,1} (A002937) But I don't recall the phenomenon in sextics. (Is H. M. Stark, ‘An explanation of some exotic continued fractions found by Brillhart, in A. O. L. Atkin and B. J. Birch eds., Computers in Number Theory, Academic Press, London and New York, 1971 available online? Hmm, √163.) NDE>No septics here. This family of octics factors over quadratic extensions, so it's "just" solving a quartic over Q(sqrt(d)). Yow, it Gaussian-factors! In[338]:= Factor[4 x^8 + 8 x + 13, Extension -> I] Out[338]= ((3 - 2 I) + (4 + 4 I) x - (4 - 2 I) x^2 - 4 I x^3 + 2 x^4) * ((3 + 2 I) + (4 - 4 I) x - (4 + 2 I) x^2 + 4 I x^3 + 2 x^4) --rwg