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Then we have ab π1ab Spec(Ok ) = Gal k˜+ /k . 10), the fundamental map ρk induces the isomorphism Jk /k × (kv× )2 × v∈Sk∞ Op× ab Gal k˜+ /k . p∈X0 Note that the left-hand side is isomorphic to the narrow ideal class group H + (k) by the correspondence Jk (av ) → p∈X0 pvp (ap ) ∈ Ik . Therefore, we have the following canonical isomorphism: H + (k) ab Gal k˜+ /k . Let k˜ ab be the maximal Abelian extension such that any prime of k is unramified, called the Hilbert class field of k. Then the Galois group Gal(k˜ ab /k) is canonically isomorphic to the ideal class group H (k) of k: H (k) Gal k˜ ab /k .

The integer ei is called the ramification index of Pi in K/F . We say that p is unramified in K/F if e1 = · · · = er = 1, and we say that p is ramified in K/F if ei > 1 for some i. We say that p is totally ramified in K/F if r = 1, e1 = n, and we say p is completely decomposed in K/F if r = n, e1 = · · · = er = 1. We also say that p is inert in K/F if r = e1 = · · · = er = 1. If any p ∈ Max(A) is unramified in K/F , K/F is called an unramified extension, and if there is a p ∈ Max(A) which is ramified K/F , K/F is called a ramified extension.

Set On := Okn , Mn := Spec(On ) and Xn := Spec(On [ p1 ]) for simplicity. 1), the natural map Mn → M0 = Spec(Z) is a Galois covering ramified over (p), and Xn → X0 = Spec(Z[ p1 ]) is the associated étale covering. The Galois group is given by Gal(Mn /M0 ) = Gal(Xn /X0 ) = Gal(kn /k0 ) Z/p n Z × . By the natural maps Mn+1 → Mn and Xn+1 → Xn , M∞ := limn Mn is a pro-finite ← − ramified Galois covering over M0 and X∞ := limn Xn is a pro-finite Galois covering ← − over X0 . Let k∞ := limn kn = Q(ζpn | n ≥ 1).

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