Some Elements of Finite Order in K 2â„š

Cheng, Xiao Yun; Xia, Jian Guo; Qin, Hou Rong
May 2007
Acta Mathematica Sinica;May2007, Vol. 23 Issue 5, p819
Academic Journal
Let K 2 be the Milnor functor and let Φ n ( x) ∈ ℚ[ x] be the n-th cyclotomic polynomial. Let G n (ℚ) denote a subset consisting of elements of the form { a,Φ n ( a)}, where a ∈ ℚ*. and {, } denotes the Steinberg symbol in K 2ℚ. J. Browkin proved that G n(ℚ) is a subgroup of K 2ℚ if n = 1, 2, 3, 4 or 6 and conjectured that G n (ℚ) is not a group for any other values of n. This conjecture was confirmed for n = 2 r 3 s or n = p r , where p ≥ 5 is a prime number such that h(ℚ(ζ p )) is not divisible by p. In this paper we confirm the conjecture for some n, where n is not of the above forms, more precisely, for n = 15, 21, 33, 35, 60 or 105.


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