J. Math. Soc. Japan Vol. 20, Nos. 1-2, 1968 on explicit formulas for the norm residue symbol Dedicated to Professor Shokichi Iyanaga on his 60th birthday By Kenkichi IWASAWA*) (Received Dec. 1, 1966) Let p be an odd prime and let Zp and % denote the ring of p-adic integers and the field of p-adic numbers respectively. For each integer n >_0, let qn = pn+1 and let l n denote the local cyclotomic field of qn-th roots of unity over Qp. We fix a primitive qn-th root of unity ~n in 'P,z so that ~n+l=fin, and put ~n =1-~n ; in generates the unique prime ideal n in the ring on of local integers in Let (a, ,3)n denote Hilbert's norm residue symbol in CD,,for the power qn and let (a, P)n =C~,a]n-- n with [a, j3]n in Zp, well determined mod qn. The classical formulas for [a, j3], state that [, ,3]n= gn1Tn(log9), j3 E l+~n? n >_0, [7n, j3], _ -gn1Tn(~n7r;1log j3), E 1+~n, n ~ 0, [a, ~]o = -go 1To(~oa~i dudao log ~) , a 1+~0 where Tn denotes the trace from to to Qpl). In a previous note [7], we have announced formulas for [a, ~3]n which generalize the above formulas of Artin-Hasse. In the present paper, we shall prove those formulas and then discuss some related results. As in the above, we retain most of the notations introduced in our earlier paper [6]. In particular, we denote by N72the norm from ~n to Qp, and by Tn,m and Nn,m the trace and the norm, respectively, from P,,. to ~n, n2 >_n >_0 ; for an automorphism a of the union P of all ~,t, n ? 0, we denote by ic(a) the *) The present research was supported in part by the National Science Foundation grant NSF-GP-4361. 1) See [1], [2], [5]. For the general theory of the norm residue symbol needed in the following, see [2], Chap. 12 or [4], II, § 11, § 19. It is to be noted that the symbol (a, j3)nin [2] is the inverse of the same symbol in [4]. Here we follow the definition of (a, IS),,in [2] as we did in [7]. 152 K. IWASAWA unique p-adic unit such that ~n = ~n(o)for every n >_02>. 1. We shall first prove several elementary lemmas. For n >_0, we denote by vn the normalized valuation on ~n such that i(r) =1. LEMMA1. Let bn be the different of ~nlQp : bn = gnp;pn• Then Tn(bnlog (1+)) 0 mod q. PROOF. Since the multiplicative group (1+pn)/(1+pn) is generated by the coset of we have log (1+pn)= log (1+pn). We shall show that Tn(bnlog a)~0 mod q,, or, equivalently, vn(gnlog a) >_2pn, for any a in 1+pn. Let a =1---3 with j3 in p, so that log a = - i-rR2 It is sufficient to show vn(gni-1Jai) >_ 2pn 2=1 - for i >_1. Let pe (e >_0) be the exact power of p dividing i. Then vn(gni-rj3z) >_(n-F1-e)(p-1)pn+2i >_(n+l_e)(p-1)pn+2pe. If e <_n, then clearly (n+1-e) (p-1)pn+2pe (p-1)pn ? 2pn. If e > n, then (n+l-e)(p-1)pn+2pe= (2-~2(pe-n -1)--(e--n-1)(p-1))pn ? 2pn because pe-n_1= (1+p-1)m-1 > (e- n)(p-1). Thus v,z(gni-1/32)>_ 2pn in either case. LEMMA2. i) qmTm(bm log (1+in)) . 0 mod q,, m >_2n+1, ii) qmTm(bm log (1+pnn)) -= 0 mod qn , m >_n+1, iii) qmTm(bm log (1+pnpn)) 0 mod qn , m >_n. PROOF. Clearly Tn,m(bm) = Tn,m(gmpo 1) is contained in p m-ngmpo 1, and if m >_2n+1, the latter is contained in pn+igmpo1= gmbn• Hence, for m >_2n+1, qmTm(bm log (1+p?z))= qmTn(Tn,m(bm) log (1+pn)) is contained in Tn(bnlog (1+p,)), and i) follows from Lemma 1. Since (1+pnn)/(1+pnn+x) is generated by the coset of ~o, we have log (1+ppn) log (l+1) = pnn~-r. Hence qmTm(bm log (l+~nn)) = Tm(pn)- 0 mod pm, and ii) holds for m >_n+1. iii) can be proved similarly by using log (1+pnpn)= pnpn po and Tfpo) = 0 mod q,. Let A denote the ring of formal power series in T with coefficients in Zp : A = ZP[[T ]]. For any nonzero element in on, there exists a power series f (T) in A such that e =f(irm), f(T) _ a1Ti , ai Zp, s = yn(e)> 0. 2=S Such f (T) will be simply called a power series for e. Let 2) However, (a, f3)n in the formulas of [6], § 3, is the inverse of the same symbol 1n the present paper. See the footnote 1). Explicit formulas for the norm residue symbol 153 d7rn n ~- din where f'(T) denotes the formal derivative of f (T) with respect to T. Since a power series for such as f (T) is not unique for , de/d r and on(e) are not uniquely determined by . However we can prove the following lemma on the values taken by on(e). LEMMA3. The values of 6)() which are obtained from all possible power series for fulfil a residue class of 1'nlmod bn. PROOF. It is clear from the definition that C (n) always belongs to pn1. In general, let f (T) and g(T) be power series in A such that f (2v,)= g(7rn). Then f(T) = g(T)+u(T)d(T ), p-1 where u(T) is a certain power series in A and d(T) _ (1-T )Zpn is the minimal z=o polynomial of 7r= 1--~n over Q~3). Differentiating the both sides, we obtain f'(in) = g'(in)+ u(in)d'(r ) Since bn = (d'(7rn)), it follows that f'(in) = g'(7r) mod u(2r)b. Now, suppose that both f (T) and g(T) are power series for . Then the co- efficient of T i vanishes for 0 < i < s ^ vn() in both f (T) and g (T ). Since d(0) = p ~ 0, the same holds for the coefficient of T z in u(T) for 0 < i < s. Hence vn(u(1rn)) > s and it follows from the above congruence that -'g'C r ,) mod bn . On the other hand, let be any element of bn = (d'(Trn)). Then there is a power series v(T) in A such that _ ~nv(7rn)d'(7rn). Let h(T) _ (1 + v(T)d(T ))f (T ) Then h(T) is also a power series for and ~n -xh'(icn) = bne-lf'@rn)+Y~. There- fore the lemma is proved. In the following, we regard on(e) as a multi-valued function of e 0 in cn, representing any such value as described above. LEMMA 4. i) For 0 and ~ 0 in o, on(e)+on(rl) mod bn . ii) For any 0 in on and for any automorphism o of D, 3) See [2], p. 151. 154 K. TWASAWA on(r) _ A:(1)on(E)' mod bn . iii) For m >_ n and for any unit in on, om() = p m-non(e) mod bm. PROOF. i) follows immediately from the fact that if f (T) is a power series. for and g(T) a power series for a~, then f (T)g(T) is a power series for r~ To prove ii), let proveToii), let u(T)=1-(1-T)~(' _ Then u(T) is a power series for 'r and u'(rn) _ (~}(1-7cn)~(~~-1= icCr)~n^1 Let f (T) be a power series for . Then f (u(T )) is a power series for r, and on(r) = bn'f'(u(in))u'(irn) -' bn(-1)df'(~n)~~(~)~n 1 =i(a)(~n-lf'(7n))d _ Finally, let be a unit in on. Then is also a unit in Dm and every power series g(T) such that g(7rm) _ is a power series for e in Urn. Let f(T) be a power series for in on. Since f (2r) _ , 7Cn=1-(1- 7rm)pm-n,f(1-(1_T )pm-n) is a power series for in Dm. Computing de/d7r?n by means of this power series, we obtain immediately the formula in iii). Note that the both sides of the congruence in iii) are well determined mod bm because bm= pm-nbn• Let now be an arbitrary element of the multiplicative group Dn of ~nr i. e., an arbitrary nonzero element of ~n. We write in the form e _ with 1 ~ 0, e2 * 0 In o,, and define ~n() - ~nCl) Un(2) By Lemma 4, i), we see that the values of on(e) again fulfil a residue class of pnl mod bn. Furthermore, i), ii) of Lemma 4 now hold for any and in Pn Hence on defines a so-called ic-homomorphism o,: ~n -;1/b. n o , is continuous in the sense that if .1 mod pn , k >>0, then on(e) = 0 mod bn. LEMMA 5. For m >_ n and for in Pm , on(Nn,m(e)) = p (m-n)Tn,m(om(e)) mod bn .