Context :

Aims : 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 ^ { \circ } , b = -6 ^ { \circ } , and b = -12 ^ { \circ } .
A fourth field including NGC6553 is also included in the discussion .

Methods : About 800 bulge field K giants were observed with the GIRAFFE spectrograph of FLAMES @ VLT at spectral resolution R \sim 20,000 .
Several of them were observed again with UVES at R \sim 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 .

Results : The IDF derived here is centered on solar metallicity , and extends from [ Fe/H ] \sim -1.5 to [ Fe/H ] \sim +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 ^ { \circ } and b = -6 ^ { \circ } ( [ Fe/H ] decreasing \sim by 0.6 dex per kpc ) .
The field at b = -12 ^ { \circ } is consistent with the presence of a gradient , but its quantification is complicated by the higher disk/bulge fraction in this field .

Conclusions : 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 .