Using a chirally invariant effective Lagrangian, we calculate the density and isospin dependences of the in-medium axial coupling, g * A , in spatially uniform matter present in core collapse supernovae and neutron stars. The quenching of g * A with density in matter with different proton fractions is found to be similar. However, our results suggest that the quenching of the nucleon's g * A in matter with hyperons is likely to be significantly greater than in matter with nucleons only. PACS: 97.60.Jd, 21.65.+f, 12.39.Fe, 26.60.+c The accurately measured beta decay lifetime of the neutron in vacuum, n → p + e − +ν e , fixes the ratio of the axial and vector couplings of the neutron to be |g A /g V | = 1.2601 ± 0.0025 [1]. Studies of beta decays in nuclei, however, have long suggested that a value of |g A /g V | ≃ 1 better fits the observed systematics [2]. Such a lower value also appears to be consistent with pion-nucleus optical potentials [3] and the systematics of Gamow-Teller resonances in nuclei [4]. Data from muon capture on nuclei, in which the relevant momentum transfer q 2 ≃ −0.9m 2 µ , have been recently analyzed including detailed nuclear structure effects [5] with the conclusion that a quenched g A (assuming g V ∼ = 1) is not necessary inasmuch as the vacuum value of g A adequately accounts for the data.The above experiments measure space-like axial transitions in nuclei. At finite density, however, space-like and time-like axial matrix elements are not necessarily equal, since Lorentz invariance is broken. In fact, there are indications from experiments with first-forbidden β decays of light nuclei that the time-like axial charge increases by about 25% in medium [6]. A theoretical expectation of this enhancement in terms of soft-pion exchanges has been offered in Ref. [7].The space-like quenching of g A in nuclei at low momentum transfers has been attributed to a combination of effects, including the partial restoration of chiral symmetry in a nuclear medium, the direct participation of the ∆(1232) in renormalizing the in-medium axial-vector current, and tensor correlations in nuclei in which shell structure effects are important [3]. An illuminating discussion of the extent to which the quenching phenomenon is intrinsic to the basic property of the "vacuum" defined by a baryon-rich medium has been given by Rho (cf. Ref. [8] and references therein). Later discussions of the quenching phenomenon in chiral approaches to spatially uniform matter can be found in Ref. [9]. The issue of breaking and restoring fundamental symmetries at large baryon density is presently intractable in lattice gauge simulations. We therefore employ an effective field-theoretical approach, based upon chiral symmetry, to consider medium modifications of the nucleon's axial coupling.The precise value of the in-meidum axial coupling, denoted by g * A hereafter, in the dense matter encountered in astrophysical phenomena such as core collapse supernovae and neutron stars is crucially important. In these cases, weak interactio...