We present the analysis of the FLAMES dataset targeting the central 25 arcmin region of the Sextans dSph. This dataset is the third major part of the high resolution spectroscopic section of the ESO large program 171.B-0588(A) obtained by the Dwarf galaxy Abundances and Radial-velocities Team (DART). Our sample is composed of red giant branch stars down to the level of the horizontal branch in Sextans. It allows to address questions related to both stellar nucleosynthesis and galaxy evolution. We provide metallicities for 81 stars, which cover the wide [Fe/H]=$-$3.2 to $-$1.5 dex range. The abundances of 10 other elements are derived: Mg, Ca, Ti, Sc, Cr, Mn, Co, Ni, Ba, and Eu. Despite its small mass, Sextans is a chemically evolved system, with evidence for the contribution of core-collapse and Type Ia supernovae as well as low metallicity AGBs. This new FLAMES sample offers a sufficiently large number of stars with chemical abundances derived at high accuracy to firmly establish the existence of a plateau in [$alpha$/Fe] at $sim 0.4$ dex, followed by a decrease above [Fe/H]$sim-2$ dex. This is in stark similarity with the Fornax and Sculptor dSphs despite their very different masses and star formation histories. This suggests that these three galaxies had very similar star formation efficiencies in their early formation phases, probably driven by the early accretion of smaller galactic fragments, until the UV-background heating impacted them in different ways. The parallel between the Sculptor and Sextans dSph is also striking when considering Ba and Eu. Finally, as to the iron-peak elements, the decline of [Co/Fe] and [Ni/Fe] above [Fe/H]$sim -2$ implies that the production yields of Ni and Co in SNeIa is lower than that of Fe. The decrease in [Ni/Fe] favours models of SNeIa based on the explosion of double degenerate sub-Chandrasekhar mass white dwarfs.
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