On Wed, Dec 26, 2018 at 8:38 PM Elkies, Noam <elkies@math.harvard.edu> wrote:
Bill Gosper <billgosper@gmail.com> writes:
[ . . . ] Do these (often dreadful) algebraic factors always come out in radicals? ---rwg
There's an extensive theory of "complex multiplication" that shows (among many other things) that if a,b are rational numbers with b>0 then DedekindEta(a+b*I) / DedekindEta(I) is algebraic, and that some power of it (at most the 24th, but maybe one never has to go that far) generates an abelian extension of the Gaussian numbers.
So THAT's what Hilbert was raving about! That means that there's always
an expression "in radicals", but you probably don't want to write it that way past the first few cases. You don't even want to see a solution in radicals of x^5 - 2*x^4 + 2*x^3 - x^2 + 1 . . .
NDE
polySolve[x^5 - 2*x^4 + 2*x^3 - x^2 + 1] {{x -> 2/5 + (-597 - 215 Sqrt[5] + 5 Sqrt[94 (265 + 101 Sqrt[5])])^( 1/5)/(5 2^( 2/5)) + ((-597 - 215 Sqrt[5] - 5 Sqrt[94 (265 + 101 Sqrt[5])])^( 1/5) E^(-((2 I \[Pi])/5)))/( 5 2^(2/5)) + ((-597 + 215 Sqrt[5] - 5 Sqrt[94 (265 - 101 Sqrt[5])])^(1/5) E^((2 I \[Pi])/5))/( 5 2^(2/5)) + ((-597 + 215 Sqrt[5] + 5 Sqrt[94 (265 - 101 Sqrt[5])])^(1/5) E^((2 I \[Pi])/5))/(5 2^(2/5))}, {x -> (2/5) + . . .}}, {x -> (2/5) + . . .}}, {x -> (2/5) + . . .}}, {x -> (2/5) + . . .}}} (somehow turning 1s and 2s into 265 and 597.-) Actually, for 1 + 110287494 x + 70410440943 x^2 - 83149258860 x^3 + 380834340687 x^4 - 313810596090 x^5 + 282429536481 x^6, the sextic behind η(i+⅓), polysolve returned an expression so large as to defeat even one step of simplification. Many thanks for all of the above. —rwg In[1]:= MinimalPolynomial[ 2/5 + (-597 - 215 Sqrt[5] + 5 Sqrt[94 (265 + 101 Sqrt[5])])^(1/5)/( 5 2^(2/5)) + ((-597 - 215 Sqrt[5] - 5 Sqrt[94 (265 + 101 Sqrt[5])])^(1/5) E^(-((2 I \[Pi])/5)))/( 5 2^(2/5)) + ((-597 + 215 Sqrt[5] - 5 Sqrt[94 (265 - 101 Sqrt[5])])^(1/5) E^((2 I \[Pi])/5))/( 5 2^(2/5)) + ((-597 + 215 Sqrt[5] + 5 Sqrt[94 (265 - 101 Sqrt[5])])^(1/5) E^((2 I \[Pi])/5))/( 5 2^(2/5))] // Timing Out[1]= {12.258655, 1 - #1^2 + 2 #1^3 - 2 #1^4 + #1^5 &}