We determine the iron distribution function (IDF) for bulge field stars, in three different fields along the Galactic minor axis and at latitudes b=-4 deg, b=-6 deg, and b=-12 deg. A fourth field including NGC6553 is also included in the discussion. About 800 bulge field K giants were observed with the GIRAFFE spectrograph of FLAMES@VLT at spectral resolution R~20,000. Several of them were observed again with UVES at R~45,000 to insure the accuracy of the measurements. The LTE abundance analysis yielded stellar parameters and iron abundances that allowed us to construct an IDF for the bulge that, for the first time, is based on high-resolution spectroscopy for each individual star. The IDF derived here is centered on solar metallicity, and extends from [Fe/H]~ -1.5 to [Fe/H]~ +0.5. The distribution is asymmetric, with a sharper cutoff on the high-metallicity side, and it is narrower than previously measured. A variation in the mean metallicity along the bulge minor axis is clearly between b=-4 deg and b=-6 deg ([Fe/H] decreasing by ~ 0.6 dex per kpc). The field at b=-12 deg is consistent with the presence of a gradient, but its quantification is complicated by the higher disk/bulge fraction in this field. Our findings support a scenario in which both infall and outflow were important during the bulge formation, and then suggest the presence of a radial gradient, which poses some challenges to the scenario in which the bulge would result solely from the vertical heating of the bar.
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