Doubt with Ampere and Biot-Savart

You didn't do the Biot-Savart integral right. Let the wire go up the z-axis, and the integral is

[tex]B(\rho) = \int_{-\infty}^\infty \frac{\mu_0}{4\pi} \frac{I e_z \; dz' \times (\rho e_\rho - z' e_z)}{(\rho^2 + {z'}^2)^{3/2}}[/tex]

You treated the denominator like it doesn't depend on [itex]z'[/itex], but it does. I write it [itex]z'[/itex] to emphasize that it is the integration variable (not the position we want to find the magnetic field at). Consult a table of integrals to easily find the antiderivative.

In general, for some current density [itex]j[/itex], the Biot-Savart Law is

[tex]B(r) = \int_{V'} \mu_0 j(r') \; dV' \times \frac{r - r'}{4 \pi |r - r'|^3}[/tex]

Actual line currents (not densities) just reduce this integral from a volume to a line. Compare with the electric field from some charge density:

[tex]E(r) = \int_{V'} \frac{\rho(r')}{\epsilon_0} \; dV' \frac{r - r'}{4 \pi |r - r'|^3}[/tex]

for vectors [itex]r, r'[/itex]. You can see these are both really the "same" law. The function [itex](r-r')/4\pi|r-r'|^3[/itex] has special significance in 3D space. Wiki "Green's functions" if you're interested in learning about it.