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I am having a bit of a problem with this classic example. I know about the symmetry but I would like to push through the math anyway. The field point is at distance Z from loop on Z axis. Using cylindrical coordinates I am at:

dH = I*(rho)d(phi)*a(phi) X (z*az - rho*a(rho)) / 4*pi*((rho)^2 + z^2))(3/2)

I know H(rho) doesn't contribute to total H. If I remove rho*a(rho) I have a cross product of az and a(phi) which is a(rho). I don't want to just jump to the numerator being i*(rho)^2*az or use trig substitutions so where is my mistake?