We discuss some results from a recent set of magnetohydrodynamic simulations of the formation of the first stars, including cosmological initial conditions and a full treatment of primordial chemistry. These simulations follow the exponential amplification of seed magnetic fields by small-scale dynamo action, and we explore some of the details of the turbulence that leads to this behavior. The key to understanding the growth of magnetic fields is the growth of vorticity, and we focus on its generation and dissipation in simulations of varying resolution. We show that the production and amplification of vorticity changes with increasing resolution, leading to the previous observations of dynamo growth rates and late time field strengths increasing with increased resolution.