Idempotents and divisible subgroups Sample Clauses

Idempotents and divisible subgroups.Β For a finite group Ξ“ we write 𝑒Γ for the idempotent |Ξ“|–1 𝛾 Ξ“ 𝛾 of Q[Ξ“]. Now fix a field 𝐿 e Ξ© and define an idempotent of Q[𝐿+ ] by setting 𝑒𝐿 := ( . 1 if 𝔣(𝐿) is a prime power β„“ |𝔣(𝐿) (1 – 𝑒𝐷𝐿,β„“ ) otherwise + [ + ]𝐿 𝐿 𝐿 ( ) where 𝔣𝐿 is the conductor of 𝐿, in all cases the products run over prime divisors β„“ of 𝔣 𝐿 and 𝐷𝐿,β„“ is the decomposition subgroup of  with respect to β„“. We also write 𝐼 for the Z  -annihilator of the element πœ‚cyc,1+𝜏 of π‘ˆ j . 𝐿 𝐿+ |𝐿 |–1 𝑔 e𝐿 πœ’ (𝑔)𝑔–1 of Q[𝐿]. For .every character πœ’ e ξˆ³Λ†πΏ we finally write π‘’πœ’ for the primitive orthogonal idempotent The following Xxxxx describes the link between the idempotent 𝑒𝐿 and the ideal 𝐼𝐿: Lemma 3.3.11.
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