General notation for cyclotomic fields Sample Clauses

General notation for cyclotomic fields. We first fix some general notation concerning cyclotomic fields. For each natural number n we fix a primitive pn − th root of unity ζpn in Qc with the pn property that ζp = ζpn−1 for all n > 1. For each natural number n we set Ln := Q(ζpn ), Gn := Gal(Ln/ Q) and Pn := Gal(Ln/L1). We also write Hn for the unique subgroup of Gn of order p − 1 and use the restriction map Gn → G1 to identify Hn with G1. p For each finite abelian group Γ we set Γ∗ := Hom(Γ, Qc×) and for each φ in Γ∗ we define p an idempotent of Qc [Γ] by setting Σ eφ := |Γ|—1 φ(γ)γ—1. γ∈Γ We recall that, by ‘orthogonality of characters’ one has eφ · eφ′ = 0 of φ /= φr and that the sum of eφ over all φ in Γ∗ is equal to 1 ∈ Zp[Γ]. We write 1Γ for the trivial element of Γ∗ and often abbreviate the idempotent e1Γ to eΓ. For each element x of Cp[Γ] and each φ in Γ∗ we write xφ for the unique element of Cp that is defined by the equality Σ x = xφeφ. (7.1) φ∈Γ∗ We use the fact that the natural direct product decomposition of abelian groups Gn = Pn × Hn implies that each element θ of G can be written uniquely as a product θ = ψ × φ (usually written as θ = ψφ) with ψ in Pn∗ and φ in Hn∗ and also that for each φ in Hn∗ the idempotent eφ belongs to Zp[Hn]. We write τn for the (unique) complex conjugation in Hn and define idempotents e+ := (1 + τn)/2 and e— := (1 − τn)/2 of Zp[Gn]. for any Zp[Gn]-module M we use submodules M + := e+(M ) and M— := e—(M ) upon which τn acts as multiplication by +1 and −1 respectively. In particular, E+ denotes the maximal real subfield of each subfield E of Ln. We also often use the fact that any Zp[Gn]-module M has a natural direct sum decomposition M = M + ⊕ M—. n We write G∗,— n and G∗,+ for the subsets of G∗n comprising characters that are odd (i.e. n with χ(τ ) = −1) and even (χ(τ ) = 1) respectively. We define subsets H∗,+ and H∗,— of H∗ n n n n in a similar way and write ω for the Teichmuller character in H∗,— = G∗,—.