Proof. Now suppose that ρp is absolutely reducible and we want to show that for any other λ ∈ Spec(E), ρλ is also absolutely reducible. Note that there exists a finite extension K over Ep such that ρp is reducible. Then there is a vector e1 in the underline space V ′ = V ⊗Ep K such that G is stable over e1. Let χ1 be the character of G acts on e1, χ2 the character G acts on V ′/K · e1. Since χi...
