By John M.H. Olmsted

ISBN-10: 0130109835

ISBN-13: 9780130109835

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PROOF. We show first that p is injective. Certainly Ker(p)= n N NeA'" and so we need only demonstrate that this intersection is trivial. Let oeKer(p) and let xeK. Then by elementary field theory there exists a finite Galois extension F'IF such that F'~K and xeF'. Now the restriction map from G=Gal(KIF) to Gal(F'IF) has kernel Gal(KIF'), which is therefore a normal subgroup ofGoffinite index. But then oeGal(KIF'), and so o(x)=x. Since x is arbitrary, eris the identity on K, and Ker(p) is trivial, as required.

We claim that a has dense image in G'. Granting this, we conclude the argmnent as follows: Since G is compact 28 1. Topological Groups and G' is Hausdorff, the image of a is, moreover, closed in G'. Thus Im(a), being dense, must be all of G', as required. To establish the claim we shall show that no open subset of G' is disjoint from Im(a). /(SN)' where SN is an arbitrary subset of GIN. } (SN). Such an intersection U consists of elements of the form where at most only finitely many of the coordinates are constrained to lie in some given proper subset of the corresponding quotient; the rest are unrestricted.

The point, of course, is to show that G is isomorphic to the projective limit of this system. 1-15 LEMMA. N where Nvaries overA', as defined above. Then there exists a surjective, continuous homomorphism a: G~G'. For NeA'"let lXrv denote the canonical projection from G to GIN, which is sUijective. Since GIN is homogeneous, we establish that lXrv is also continuous by noting that a/J(eG1N ) = N, which by hypothesis is open in G. #", the following triangle is commutative: PROOF. #", where PN denotes projection from G' onto GIN, the component of the projective limit corresponding to N.

### Advanced calculus by John M.H. Olmsted

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