From the header line "quintic trivia", the resolvent quartic of the solvable quintic needs only square roots of square roots (no cube roots).
I believe it is the same with the next Fermat degree, namely the 17-deg. It's resolvent 16-deg needs only square roots of square roots of square roots, etc. Here's a "natural" solvable 17-th deg with small coefficients: -1 + x + x^2 + x^3 - x^5 - 13 x^6 - 7 x^7 - 11 x^8 - 4 x^9 - x^10 - 7 x^11 - 23 x^12 - 31 x^13 - 42 x^14 - 24 x^15 - 6 x^16 + x^17 = 0 It's 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. I was able to find it's 16-deg resolvent, but I don't have my notes here with me. :-( - Tito On Sun, Sep 4, 2011 at 4:34 PM, Bill Gosper <billgosper@gmail.com> wrote:
On Sat, Aug 6, 2011 at 3:11 PM, Bill Gosper <billgosper@gmail.com>wrote:[...]
Recall that quartic solutions in general have quadratic surds inside cubic
surds inside quartic surds,
No, Julian points out that In[642]:= Position[Solve[x^4 + c*x^2 + d*x + e == 0, x][[1]], _^(1/4)|_^(-1/4)]
Out[642]= {}
I.e. Mathematica does it with square roots of square roots of cube roots of square roots. However in Macsyma, reveal(solve(x^4+c*x^2+d*x+e,x)[1],7) Sum(4) sqrt(------) sqrt(- Quotient) Expt x = - ---------------- - ------------ 1/4 6 6 Quotient --rwg