By O. T. O’Meara (auth.)
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Using the fact that multiplication is continuous in the p-adic topology we can obtain in this way an element of o (S) that is arbitrarily close to 1 at p and to 0 at all q E T- p. Let us denote such an element by Av· Then obtain an Av for each p E T. 3) The rest is easy. Just form E ocvAv· This element satisfies rET E IXpApl ~ IPET q 1 V q Es - T by choice of T and the Av. And by continuity of addition and multiplication it can be made arbitrarily close to IXp simultaneously at all p E T just by making the approximations Av in step 2) sharp enough.
Me that ~r = 1 for all p ES - T by enlarging T if necessary. Pick IXp EF for each p E T in such a way that Jocvlr = ~v· Choose A E F with VpET { JA - ~Xvlr < I~Xvlr VpES-T. JAlv~ 1 This A is the required element. q. e. d. 21:3. LetS be a Dedekind set of spots on F. Then o(S) cF. And Fis the quotient field of o (S). Proof. Wehave o(p)CF for any p ES since all spotsinS are nontrivial. Hence o (S) cF. Consider a typical oc E F which we wish to express as a quotient of elements of o (S). Put T = {p E SJ Joclv> I}.
Conversely, suppose p is discrete. Then where e= e($I p) is the ramification index of the extension. Hence lEI must be a discrete subgroup of P. Hence $ is discrete. q. e. d. Consider a discrete spot p on F. Recall from ring theory that an element n in the integral domain o (p) is called a prime element of o (p) if it is a non-unit suchthat in every factorization n = IX ßwith IX, ß Eo (p) either IX or ß is a unit: By a prime element of F at p we shall mean a prime element of the integral domain o (p) in the above sense.