A maxima package for recognition of floats.
(%i1) load(identify)$
(%i2) root: float(sin(%pi/12))$
(%i3) identify(root);
(%o3) sqrt(2-sqrt(3))/2
(%i4) makelist(root^i, i, 0, 4)$
(%i5) integer_relation(%);
(%o5) [-1,0,16,0,-16]
(%i6) makelist(x^i, i, 0, 4).%;
(%o6) -16*x^4+16*x^2-1
(%i7) solve(%);
(%o7) [x=-sqrt(sqrt(3)+2)/2,x=sqrt(sqrt(3)+2)/2,x=-sqrt(2-sqrt(3))/2,x=sqrt(2-sqrt(3))/2]