The case |G| = p2 Sample Clauses

The case |G| = p2. In this section we fix a Galois extension of number fields L/K of degree p2, with G := Gal(L/K) and also a finite set of places Σ of K that contains all archimedean places, all places above p and all that ramify in L/K. We write F for the (unique) intermediate field of L/K with [L : F ] = p and set H := Gal(L/F ). We also set r := rK and note that rL = p2 · rK. In this case, Xxxxxx and Xxxxxx [20] have shown that there are 4p +1 isomorphism classes of indecomposable Zp[G]-lattices. The basic properties of these lattices are conveniently recorded in Table II given below (and taken from [20, Table 2]). In this table we write R1 and R2 for the modules Zp[x] and Zp[x] , where Φ i (x) is the (Φp(x)) (Φp2 (x)) p pi—th cyclotomic polynomial, and in each case a fixed generator of G acts on the quotient as multiplication by x. For a Zp[G/H]-lattice X the notation (R2, X; λi ) denotes a module that arises as the exten- Z [G]p sion of R2 by X that corresponds to the i-th power of a certain element λ0 of Ext1 (R2, X) ~= X/pX.