The case |G| = p Sample Clauses

The case |G| = p. In this section we fix a Galois extension of number fields L/K of degree p and set G := Gal(L/K). We also fix a finite set of places of K that contains all archimedean places, all that ramify in L/K and all that divide p. In this case there are three isomorphism classes of indecomposable Zp[G]-lattices and there basic properties are given by the following table (taken from [35]) in which we write TrG for the element Σg∈G g of Zp[G]. Table I M H0(G, M ) Hˆ—1(G, M ) rkZ (M )p Zp[G] Zp 0 p Op := Zp[G]/TrG 0 Z /p Z p − 1 Zp Zp 0 1 By using this classification, the following result gives a complete description of the Galois structure of the Zp[G]-lattice AΣ in terms of the finite group T Σ defined at the beginning L of Chapter 4. L/K Proposition 5.1.1. Let p be an odd prime, assume that Xxxxxxxx’s Conjecture is valid for L at p and set d := dimF (T Σ ). Then there is an isomorphism of Zp[G]-modules p L/K L p p AΣ ~= Zp[G]rK—d ⊕ Od ⊕ Zd+1.