By Franz Halter-Koch

"Provides for the 1st time a concise advent to basic and multiplicative perfect conception, legitimate for commutative earrings and monoids and awarded within the language of perfect platforms on (commutative) monoids."

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Hence K' will satisfy the requirements of our Proposition. We let Cr denote a primitive r-th root of unity. 3 RECIPROCITY LAW 47 Suppose F is odd. The field is a compositum of bhe field Q(C4) and a (1—1) over At a finite has has degree (F— 1) over the completion of prime p this implies that By a local result (left to reader) it follows that the degree Q,,} — oo as n cc. Since this degree is a power oft, we can find an it such that this degree is divisible by P at a given finite set of finite primes, as desired.

Let n be a pitne. Let k(ç/&) = K be any given Kttmmer extension of degree n. 4 = kD1, and for which the associated field K1 happens to be K. Before proving this lemma we indicate right away how the second inequality follows from it. Since 4 = k*Di C N2J2) = 1. By . N2J2 we have (4: the first inequality this implies that "2 = k. The last line of Lemma 2 reads now k*N1J1) ( [K1 : k] and this is the second inequality since K = K1. ) . (4: PaooF OF LEMMA 3. Let be a finite set of primes containing • allp I n and all archimedean primes, • allpIa(whichmakesaan Si-unit), • enough primes so as to have 1k = k4'.

Let it be a prime. If n = p, the proff of the second inequality *c above) shows that, under the isomorphism of Theorem 6, a universal norm corresponds to an element of the Galois group which leaves fixed every cyclic extension of degree p, and is therefore a p-th power. We may therefore assume that it $ p. This puts us In the situation of Section 3. Then D1 = We go back to Lemma 2. We let 52 be the empty set, S = x flpqsUp = Js. Hence A1 = k5 and K1 = We also have = J. 12 it follows from the first inequality that = fl k and hence A2 = K2 = k.