Local-density calculations of the electronic structure as a function of temperature T and applied magnetic field H are presented for FeSi. The small band gap (6 mRy) makes FeSi change from being semiconducting at low T to being metallic at moderate T, and the exchange enhancement shows strong variations with T and H. An unusual situation, when a local field has closed the gap, is favorable for additional enhancement and spin fluctuations. Low-T properties are understood from mean-field calculations based on the band structure, but spin fluctuations have to be invoked to describe properties for T larger than about 200 K.