GRB 131014A: a Laboratory to Study the Thermal-Like and Non-Thermal Emissions in Gamma-Ray Bursts, and the new L$_\mathrm{i}^\mathrm{nTh}$-E$_\mathrm{peak,i}^\mathrm{nTh,rest}$ relation
Evidence has been accumulated on the existence of a thermal-like component during the prompt phase of GRBs. This component, often associated with the GRB jet's photosphere, is usually subdominant compared to a much stronger non-thermal one. The prompt emission of Fermi GRB 131014A provides a unique opportunity to study this thermal-like component. Indeed, the thermal emission in GRB 131014A is much more intense than in other GRBs and a pure thermal episode is observed during the initial 0.16 s. The thermal-like component cools monotonically during the first second while the non-thermal emission kicks off. The intensity of the non-thermal component progressively increases until being energetically dominant at late time. This is a perfect scenario to disentangle the thermal component from the non-thermal one. A low-energy spectral index of +0.6 better fit the thermal component than the typical index value +1 corresponding to a pure Planck function. The non-thermal component is adequately fitted with a Band function whose low and high energy power law indices are ~-0.7 and <~-3, respectively; this is also statistically equivalent to a cutoff power law with a ~-0.7 index. This is in agreement with our previous results. Finally, a strong correlation is observed between the time-resolved luminosity of the non-thermal component, L$_i^{nTh}$, and its corresponding rest frame spectral peak energy, E$_{peak,i}^{rest,nTh}$, with a slope similar to the one reported in our previous articles. Assuming this relation to be universal for all GRBs we estimate a redshift of ~1.55 for GRB 131014A that is a typical value for long GRBs. These observational results are consistent with the models in which the non-thermal emission is produced well above the GRB jet photosphere but they may also be compatible with other scenarios (e.g., dissipative photosphere) that are not discussed in this article.