Gamma-ray bursts are the most energetic electromagnetic sources in the Universe. Their prompt gamma-ray radiation, lasting between a fraction of a second to several thousand seconds, corresponds to an energy release of 10 42 -10 47 J [1,2]. Fifty years after their discovery and several dedicated space-based instruments, the physical origin of this emission is still unknown. Synchrotron emission has been one of the early contenders 3,4 , but was criticized because spectral fits of empirical models (such as a smoothly-connected broken power law or a cut-off power law) suggest too hard a slope of the low-energy power law, violating the so-called synchrotron line-of-death 5,6 , reviving models of photospheric emission 7-9 . Fitting proper synchrotron spectra 10-12 (rather than heuristic functions) was first shown to work for individual GRBs 13,14 , though without tracking electron cooling. When the latter was taken into account, several GRB spectra could be fit successfully 10 . Here we show that idealized synchrotron emission, when properly incorporating time-dependent cooling of the electrons, is capable of fitting~95% of all time-resolved spectra of single-peaked GRBs as measured with Fermi/GBM. The comparison with spectral fit results based on previous empirical models demonstrates that the past exclusion of synchrotron radiation as an emission mechanism derived via the line-of-death was misleading. Our analysis probes the physics of these ultra-relativistic outflows and the microphysical processes which cause them to shine, and for the first time provides estimates of magnetic field strength and Lorentz factors of the emitting region directly from spectral fits. The parameter distributions that we find are grossly compatible with theoretical spectral 15-17 and outflow 18 predictions. The emission energetics remain challenging for all theoretical models. As synchrotron radiation alone can explain the observed emission, it is difficult to reconcile the time scales, efficiencies, and microphysics predicted by relativistic Fermi shock acceleration 19 and the fireball model 20 with the observations. Thus, our modeling of the Fermi/GBM observations provides evidence that GRBs are produced by moderately magnetized jets in which relativistic mini-jets emit optically-thin synchrotron radiation at large emission radii.We perform time-resolved spectral analysis of the prompt spectra of gamma-ray bursts (GRBs) detected with the Fermi Gamma-ray Burst Monitor (GBM). We select a subset of GRBs which exhibit a single contiguous, pulse-like structure which is justified by the assumption that the emission originates from a single dissipation episode 10 . In addition, we require that all GRBs have a measured redshift to ascertain their energetics. This results in a sample of 19 out of the 81 GRBs in the GBM time-resolved spectral catalog 21 . After background modeling, we employ the Bayesian blocks algorithm to create spectral datasets in optimised temporal bins for each of those GRBs, leading to a total of 162 bins, each with a P...