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We formulate and solve the equations governing the dynamics of a microscopic artificial swimmer composed of a head and of a tail made of a thin film of permanent magnetic material. This is a variant of the model swimmer proposed by Dreyfus et al. in 2005, whose tail is a filament obtained from the assembly of superparamagnetic beads. The swimmer is actuated by an oscillating magnetic field, and its geometry is inspired by that of sperm cells. Using values for the geometric and material parameters that are realistic for a magnetic multilayer, we show that the model swimmer can reach swimming speeds exceeding one body length per second, under reasonable values of the driving magnetic field. This provides a proof of principle for the viability of the concept. In addition, we discuss the possibility to steer the system along curved paths. Finally, we compare the propulsion mechanism (swimming “gait”) of our …