Using a suite of hydrodynamical simulations with cold dark matter, baryons, and neutrinos, we present a detailed study of the effect of massive neutrinos on the 1-D and 3-D flux power spectra of the Lyman-$alpha$ (Ly$alpha$) forest. The presence of massive neutrinos in cosmology induces a scale- and time-dependent suppression of structure formation that is strongest on small scales. Measuring this suppression is a key method for inferring neutrino masses from cosmological data, and is one of the main goals of ongoing and future surveys like eBOSS, DES, LSST, Euclid or DESI. The clustering in the Ly$alpha$ forest traces the quasi-linear power at late times and on small scales. In combination with observations of the cosmic microwave background, the forest therefore provides some of the tightest constraints on the sum of the neutrino masses. However there is a well-known degeneracy between $Sigma m_{ u}$ and the amplitude of perturbations in the linear matter power spectrum. We study the corresponding degeneracy in the 1-D flux power spectrum of the Ly$alpha$ forest, and for the first time also study this degeneracy in the 3-D flux power spectrum. We show that the non-linear effects of massive neutrinos on the Ly$alpha$ forest, beyond the effect of linear power amplitude suppression, are negligible, and this degeneracy persists in the Ly$alpha$ forest observables to a high precision. We discuss the implications of this degeneracy for choosing parametrisations of the Ly$alpha$ forest for cosmological analysis.
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