Bill Gosper <billgosper@gmail.com> wrote:
In http://mathworld.wolfram.com/SquareDissection.html, x_2 satisfies [...]
These fancy \sqrt signs just make it hard to plug into a software package that might do this. Do you mean the polynomial with coefficients 1, 0, -2, -8+8*w, 28-20*w, -24+16*w, 6-4*w where w^2 = 2? That seems to have Galois group S_6 over Q(sqrt(2)). It satisfies an irreducible polynomial of degree 12 over Q, too large for gp's "polgalois"; but reducing modulo some primes finds factors of degrees 5,1 (for a prime above 23), degree 6 (both primes above 47) and degrees 4,2 (a prime above 71), which I think is enough to exclude all proper subgroups of S_6, let alone solvability by radicals. Also the discriminant has a huge prime factor of norm 218206319 with multiplicity 1. NDE