We investigate the possibility of measuring intergalactic magnetic fields using the dispersion measures and rotation measures of fast radio bursts. With Bayesian methods, we produce probability density functions for values of these measures. We distinguish between contributions from the intergalactic medium, the host galaxy and the local environment of the progenitor. To this end, we use constrained, magnetohydrodynamic simulations of the local Universe to compute lines-of-sight integrals from the position of the Milky Way. In particular, we differentiate between predominantly astrophysical and primordial origins of magnetic fields in the intergalactic medium. We test different possible types of host galaxies and probe different distribution functions of fast radio burst progenitor locations inside the host galaxy. Under the assumption that fast radio bursts are produced by magnetars, we use analytic predictions to account for the contribution of the local environment. We find that less than 100 fast radio bursts from magnetars in stellar-wind environments hosted by starburst dwarf galaxies at redshift $z gtrsim 0.5$ suffice to discriminate between predominantly primordial and astrophysical origins of intergalactic magnetic fields. However, this requires the contribution of the Milky Way to be removed with a precision of $approx 1 rm~rad~m^{-2}$. We show the potential existence of a subset of fast radio bursts whose rotation measure carry information on the strength of the intergalactic magnetic field and its origins.
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