We have obtained high-resolution spectroscopy of ten red giants in the Carina dwarf spheroidal (dSph) with UVES at the ESO/VLT. Here we present the abundances of O,Na,Mg,Si,Ca,Ti and Fe. By comparing the iron abundances [Fe/H] with calcium triplet (CaT) metallicities we show that the empirical CaT technique yields good agreement with the high-resolution data for [Fe/H]>-2 dex, but tends to deviate at lower metallicities. We identify two metal poor stars with iron abundances of -2.72 and -2.50 dex. These stars are found to have enhanced [alpha/Fe] ratios similar to those of stars in the Milky Way halo. However, the bulk of the Carina red giants are depleted in the [alpha/Fe] abundance ratios with respect to the Galactic halo at a given metallicity. One of our targets, with a [Fe/H] of -1.5 dex, is considerably depleted in almost all of the alpha-elements by ~0.5 dex compared to the solar values. Such a low [alpha/Fe] can be produced by stochastical fluctuations in terms of an incomplete mixing of single Type Ia and II SNe events into the ISM. Our derived element ratios are consistent with the episodic and extended SF in Carina known from its color-magnitude diagram. We find a considerable star-to-star scatter in the abundance ratios. This suggests that Carinas SF history varies with position within the galaxy, with incomplete mixing. Alternatively, the SF rate is so low that the high-mass stellar IMF is sparsely populated, as statistically expected in low-mass star clusters, leading to real scatter in the resultant mass-integrated yields. Both ideas are consistent with slow stochastic SF in dissolving associations, so that one may not speak of a single SF history at a detailed level (Abridged).
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