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My question is quite elementary.

I am wondering all irreducible abelian Galois representation of $Gal(\bar{Q}/Q)$ should be character.(i.e 1-dimensional).

I think it should be ture. But since no guarantee on the compactness of abelian Galois group, I am hesitating to apply Schur Theorem.

How to show this?

user29422
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    Every Galois group is a profinite group, and hence compact; you needn't be worried on that count. –  Sep 22 '13 at 08:03
  • It might help to think of the Galois group as a projective limit of finite groups coming from finite Galois extensions, instead of some abstract group of automorphisms of an algebraically closed field. – Scott Carnahan Sep 22 '13 at 19:58
  • For every locally compact abelian group the irreducible representations are one-dimensional. I am voting to close. See here: http://en.wikipedia.org/wiki/Class_field_theory – Marc Palm Sep 23 '13 at 09:59
  • Dear all, thanks too much your reply. All comments are enlightening! –  Sep 23 '13 at 15:04

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