* Bill Gosper <billgosper@gmail.com> [Feb 14. 2011 09:51]:
http://mathworld.wolfram.com/EllipticIntegralSingularValue.html leads off with "Abel (quoted in Whittaker and Watson 1990, p. 525) proved that whenever (K'(k))/(K(k))=(a+b sqrt(n))/(c+d sqrt(n)), where a, b, c, d, and n are integers, K(k) is a complete elliptic integral of the first kind, and K'(k)=K(sqrt(1-k^2)) is the complementary complete elliptic integral of the first kind, then the elliptic modulus k is the root of an algebraic equation with integer coefficients." (Note these are old style K(k) := EllipticK[k^2] .) E.g., choosing a:=0, b:=1, n:=5, c:=2, and d:=0, so that K'(k)/K(k) = Sqrt[5]/2, then k^2 -> 4*Sqrt[2]*(-(3491/2) - 1234*Sqrt[2] - (1561*Sqrt[5])/2 - 552*Sqrt[10] + (1/22)* Sqrt[38078 + 17029*Sqrt[5]]*(285 - 3*Sqrt[5] + 88*Sqrt[10]))
(probably undersimplified). OK, so change a:=1 to get K'/K = GoldenRatio. I cannot to save my life PSLQ a polynomial satisfied by LambdaStar[GoldenRatio^2]. (E.g., degree = 100, $MaxExtraPrecision = 69999.) Is the theorem false? Misquoted? I'm spazzing? It's true, but the polynomial is stupendous??
Same here, I can only obtain a minpoly for sqrt of any rational (which can be a square). So the mathworld statement appears to be incorrect (or incomplete). I'd be very interested what the correction might be.
So what do W&W say? "The theorem is beyond the scope of this book."! --rwg
[will be away from computers for 15 days, so cannot reply promptly]