Re: [math-fun] 3 was not Dedekind's lucky number

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. 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