1
$\begingroup$

I have been looking at J Neukirch's book on Algebraic Number Theory, he has a chapter on Abstract class field theory, in which he talks about Abstract valuation theory, this is where ive run into a problem. This is the set up.

We let $G$ be a profinite group, and assume we have a surjective continuous homomorphism $d:G \rightarrow \hat{\mathbb{Z}}$, now we let $G_{k}$ be a subgroup of $G$, so we can then restrict $d$ to $G_{k}$ and get a homomorphism $d:G_{k} \rightarrow \hat{\mathbb{Z}}$, he then defines $f_{k} = (\hat{\mathbb{Z}}:d(G_{k}))$.

Now what I dont understand is the following, he says if $f_{k}$ is finite then we get a surjective homomorphism $d_{k}=\frac{1}{f_{k}}d:G_{k} \rightarrow \hat{\mathbb{Z}}$, why is this surjective? and is there a good way to visualize this map when its applied to the absolute galois groups of local or global fields?

Thank you

  • 1
    The open subgroups of $\hat{\mathbf{Z}}$ are precisely the subgroups of the form $n\hat{\mathbf{Z}}$, and the corresponding quotient is $\mathbf{Z}/n\mathbf{Z}$. So if $d(G_k)$ has finite index then it is open in $\hat{\mathbf{Z}}$ (being compact, hence closed), which means $d(G_k)=f_k\hat{\mathbf{Z}}$.2012-10-12

0 Answers 0