On the kernel of the norm in some unramified number fields extensions

dpH (G,EL) = dpH (G,Z/pZ)− dpH (G,Z/pZ) + dpH (G,Z/pZ). It is so crucial to find an upperbound for the p-rank dpH (G,EL) when Cl(L) is trivial. In this paper, we prove results about this rank in some special cases. More precisely, we compute this p-rank when L/K is an abelian unramified (also at infinity) p-extension whose Galois group can be generated by two elements. We also exhibit an explicit basis of the p-group H(G,EL). Notations — Let K be a number field. We denote by ΣK the set finite places, Div(K) its divisor group and Cl(K) its divisor class group. To each finite place v ∈ ΣK one can associate a unique prime ideal pv of K and to each x ∈ K, there corresponds a principal divisor 〈x〉K of K. If L/K is a Galois extension of number fields, then for each v ∈ ΣK , ΣL,v denotes the subset of places w ∈ ΣL above v (for short w | v) and fv the residual degree of any w ∈ ΣL,v over K. The map eL/K : Div(K) → Div(L) is the classical extension of ideals. Let G be a finite group and M be a G-module. The norm map NG : M → M is defined by x 7→ ∏ g∈G g(x); its kernel is denoted by M [NG]. The augmentation ideal IGM = 〈 g(x) x , x ∈ M, g ∈ G 〉 is of importance. Of course,