Representations of field automorphism groups

This is a common introduction to math.RT/0101170, math.RT/0306333, math.RT/0506043, math.RT/0601028. Compared to these references there are new results including (i) a description of a separable closure of an extension of transcendence degree one of an algebraically closed field of positive characteristic; (ii) a “K"unneth formula” for the products with curves; (iii) the semi-simplicity of the module of regular forms of top degree ${\mathrm{\Omega }}_{F\mathrm{/}k}^n$.