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Is every continuous epimorphism from the absolute Galois group of $\mathbb{Q}$ to itself injective?

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    $\begingroup$ A lot of relevant material is here: "Cohomology of Number Fields" Neukirch, Schmidt, Wingberg (2013) mathi.uni-heidelberg.de/~schmidt/NSW2e In particular: (12.2.3) Corollary: all automorphisms of the absolute Galois group of Q are inner; (12.3.3) Question: is every open homomorphism between absolute Galois groups of number fields injective? $\endgroup$
    – David Lampert
    Commented Feb 16, 2015 at 17:11
  • $\begingroup$ @DavidLampert thank you very much for the reference to the more general question, but I was aware of it. $\endgroup$
    – Pablo
    Commented Feb 16, 2015 at 17:29

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