Fradkin's formulation of statistical field theory is applied to the Coulomb interacting electron gas in a magnetic field. The electrons are confined to a plane in normal three-dimensional space and also interact with the physical three-dimensional electromagnetic field. The magnetic-translation-group Ward identities are derived. By using them, it is shown that the exact electron propagator is diagonalized in the basis of the wave functions of the free electron in a magnetic field whenever the magnetic-translation-group symmetry is unbroken. The general tensor structure of the polarization operator is obtained and used to show that the Chern-Simons action always describes the Hall-effect properties of the system. A general proof of the Str\^eda formula for the Hall conductivity is presented. It follows that the coefficient of the Chern-Simons terms in the long-wavelength approximation is exactly given by this relation. Such a formula, expressing the Hall conductivity as a simple derivative, in combination with a diagonal form of the full propagator, allows us to obtain a simple expression for the filling factor and the Hall conductivity. Indeed, these results, after assuming that the chemical potential lies in a gap of the density of states, lead to the conclusion that the Hall conductivity is given without corrections by ${\mathrm{\ensuremath{\sigma}}}_{\mathit{x}\mathit{y}}$=\ensuremath{\nu}
e2/h, where \ensuremath{\nu} is the filling factor. In addition, it follows that the filling factor is independent of the magnetic field if the value of the chemical potential remains in the gap.
We present analytic formulas for the ground state energy of two-dimensional (2D) anyon gas in strong magnetic field (Landau level filling factor νL ≤ 1). The formulas are obtained by applying harmonic potential regularization for the vanishing confinement to harmonically confined Coulomb anyon gas. In a case of absence of the Coulomb interaction our analytic result provides an exact solution. It contains a contribution of the anyon gauge field characterized by the anyon parameters ν and νL. In a case of presence of the Coulomb interaction we introduce a function depending on the parameters ν, νL and the density parameter rs. The function is determined by fitting the Fano-Ortolani interpolation equation in the fractional quantum Hall effect regime for spin-polarized electrons and by consistence requirement with known results for the ground state energy of the 2D Coulomb Bose gas in strong magnetic field. We show that our formulas are valid not only for fermions (ν=1) but quite generally for anyons (0 ≤ ν ≤ 1).