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Stellar dynamos generate magnetic fields that are of fundamental importanceto the variability and evolution of Sun-like and low-mass stars, and for thedevelopment of their planetary systems. We report measurements ofsurface-average magnetic fields in 292 M dwarfs from a comparison withradiative transfer calculations; for 260 of them, this is the first measurementof this kind. Our data were obtained from more than 15,000 high-resolutionspectra taken during the CARMENES project. They reveal a relation betweenaverage field strength, <B>, and Rossby number, $Ro$, resembling thewell-studied rotation-activity relation. Among the slowly rotating stars, wefind that magnetic flux, $\Phi_\textrm{B}$, is proportional to rotation period,$P$, and among the rapidly rotating stars that average surface fields do notgrow significantly beyond the level set by the available kinetic energy.Furthermore, we find close relations between nonthermal coronal X-ray emission,chromospheric H$\alpha$ and Ca H&K emission, and magnetic flux. Taken together,these relations demonstrate empirically that the rotation-activity relation canbe traced back to a dependence of the magnetic dynamo on rotation. We advocatethe picture that the magnetic dynamo generates magnetic flux on the stellarsurface proportional to rotation rate with a saturation limit set by theavailable kinetic energy, and we provide relations for average field strengthsand nonthermal emission that are independent of the choice of the convectiveturnover time. We also find that Ca H&K emission saturates at average fieldstrengths of $\langle B \rangle \approx 800$ G while H$\alpha$ and X-rayemission grow further with stronger fields in the more rapidly rotating stars.This is in conflict with the coronal stripping scenario predicting that in themost rapidly rotating stars coronal plasma would be cooled to chromospherictemperatures.

Magnetism, rotation, and nonthermal emission in cool stars -- Average magnetic field measurements in 292 M dwarfs