Tame kernels of cubic cyclic fields

Tame kernels of quintic cyclic fields

“…In Section 4, we use the G-module structure of K 2 O F and apply reflection theorems to investigate the q-primary part of K 2 O F for odd q. In particular, we prove a theorem, similar to the main result in [15], which confirms Browkin's Conjecture 4.6 of [1]. Finally, we assume that in F there is only one ramified prime p, p > 11.…”

“…Introduction. The structure of the tame kernels of algebraic number fields has been investigated by many authors, e.g., [1], [2], [4], [5], [7], [10], [11], [12] and [15]. In particular, J. Browkin gave some explicit results for cubic cyclic fields with only one ramified prime in [1], and H. Zhou investigated the structure of tame kernels of cubic cyclic fields with two ramified primes in [15].…”

Tame kernels of cyclic extensions of number fields

“…However, even for m = 1, we do not know this value in general. In this paper we investigate the p m -rank of the tame kernel K 2 O E for a cyclic extension E/F of number fields of degree n with p n. As applications, for E/Q being a cyclic extension of odd prime order l, we obtain some results on the divisibility of p m -rank K 2 O E that generalize results for l = 3, 5 proved in [Br1], [Zh1] and [Wu]. For a cyclotomic field Q(ζ l ), we investigate the divisibility of the orders of K 2 O Q(ζ l ) for l < 2000 and l ≡ 3 (mod 4).…”

“…Some cyclic extensions of Q. Let E/Q be a cyclic extension of odd prime order l. As applications, in this section, we obtain some results on the divisibility of p m -rank K 2 O E ; they generalize results for l = 3, 5, proved in [Br1], [Zh1] and [Wu]. Using Proposition 4 below and results of [Br3], we investigate the divisibility of the orders of K 2 O Q(ζ l ) for l < 2000 and l ≡ 3 (mod 4).…”

“…Let F be an algebraic number field, O F the ring of integers in F and K 2 the Milnor K-functor. For an odd prime p, results on the p-primary part of tame kernels of number fields can be found in [Br1], [Br2], [Gu], [Ke], [Ko], [Qi], [Wu], [Zh1] and [Zh2]. For m ≥ 1, it is of interest to find the value of p m -rank K 2 O F .…”