By Dawkins P.

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The ramifications at 2 and 3 are occurring simultaneously, but we want to isolate, say, the prime 2. So we adjoin the character X3 and obtain the group generated by X2X3 and X3' which is also generated by X2 and X3. We now have the picture So we have "split" the field O(j3) so as to isolate the ramification at 2. 6. Let X be a Dirichlet character and K the associated field. ). More generally, let L be the field associated with a group X of Dirichlet characters. Then p is unramified in L/O <:> X(p) "" 0 for all X E X.

Gives another extension. ) 1 N 1 = Nf(a. ) = Nf(P rN l i N ) + Nf(l) + Nf(x ) 1 = 0 + 0 + N 10gp(xN) = logp(x). Therefore the extension is unique. l. p-adic Functions 51 If u E Gal(Op/Op) then since luxl = Ixl for all x a continuous automorphism of Cpo By continuity, logp(1 E Op we may extend u to + ux) = L (-I)n+1(ux)n/n = u L (_l)n+1 xn/n = ulogp(1 + x). , 10gp(uIX) = u(logp IX). It follows that for IX E Op, logp(IX) E Op(IX); this fact also follows from the power series expansion. For Op we may carry out the construction of the propositon more explicitly.

1 = (;;-1 ((; - 1). L/K = 1. Therefore dn = (d;;)2. If n = pa, p "# 2, then NEe = N((; - 1) = p, so d. = p(d;;)2. If n = 2a , then (; is a 2a - I st root of 1; so NEe = N((; - 1) = 4. Therefore d. = 4(d;;)2. 18 and the fact that log p = o (log n). This completes the proof. 19, we have tP(n) log d. tP(n) log dn+ ---+ 0 ' so the Brauer-Siegel theorem applies. Rn = ! 16, 10g(::) = O(tP(n». ) = t log d. - t log d,; + O(IP(n)) + o(log dn) = ilP(n)logn + o(lP(n) log n). We have proved the following result.