On Elements Of Given Order In K₂(Q) And Diophantine Equation

How to write down explicitly the elements of finite order in K2 group of a field is an important problem in the Algebraic K-theovy. J.Tate proved that if a global field F contains n a primitive n-th root of unity, then every element of order n in K2F is of the form {a, n} for some a in F*. A.A.Suslin proved that J.Tate's result is true for any field with n. For any field F, J.Browkin studied the elements of the form {a, Φn(a)} in K2F, where Φn(x) is the n-th cyclotomic polynomial. Let Gn{F) = {{a,Φn(a)}∈ K2F| a,Φn(a)∈ F*}. F2 denotes the field consisting of 2 elements. J.Browkin proved:(i) For every field F= F2 and n = 1,2,3,4 or 6 if n ∈ F, then every element {a, n} in K2F can be written in the form {b, Φn(b)} for some b ∈ F* satisfying Φn(b) = 0;(ii) For every field F = F2 and n = 1,2,3,4 or 6, Gn(F) is a subgroup of K2F.Browkin conjectured that (i) and (ii) should not hold in general for any other values of n.In my paper, I will prove G81(Q) is not a subgroup of K2(Q), which partially confirms a conjecture on K2 posed by J.Browkin in [1], and study the Diophantine equation x4 + y4 = z2 in Z[11] which may be applied to prove G2n(Q(11)) is a subgroup of K2(Q(11)) if and only if n < 2.