  We consider continuous representations of the Galois group G of a number
field K taking values in the completion C of an algebraic closure A of the
field of l-adic numbers. We give a construction of irreducible representations
of G in GL(2,C) which cannot be conjugated into GL(2,A) but for which there
exist density-one sets of primes with Frobenius traces in A. We also show that
any such representation must be ramified at infinitely many primes of K.
