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Based on the extended Mie theory with Fourier transform, the electromagnetic field in homogeneous gyrotropic media, for the first time, can be analytically obtained in spectral domain in terms of spherical eigen-vectors with their associated coefficients. The coefficients of electromagnetic fields in the gyrotropic shell and the isotropic host medium are thus exactly solved in a recursive manner. Our analytical extended Mie scattering theory have been numerically validated. Using this analytical approach, results of the scattering by the general gyrotropic coated conducting sphere are obtained.