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We present an analytic method to compute the one-loop magnetic correction tothe gluon polarization tensor starting from the Landau-level representation ofthe quark propagator in the presence of an external magnetic field. We showthat the general expression contains the vacuum contribution that can beisolated from the zero-field limit for finite gluon momentum. The generaltensor structure for the gluon polarization also contains two spurious termsthat do not satisfy the transversality properties. However, we also show thatthe coefficients of this structures vanish and thus do not contribute to thepolarization tensor, as expected. In order to check the validity of theexpressions we study the strong and weak field limits and show that wellestablished results are reproduced. The findings can be used to study theconditions for gluons to equilibrate with the magnetic field produced duringthe early stages of a relativistic heavy-ion collision.

Gluon polarization tensor in a magnetized medium: Analytic approach starting from the sum over Landau levels