We present an infinite series formula based on the Karoubi-Hamida integral, for the universal Borel class evaluated on H_{ 2n+1}(GL(\mathbb{ C})). For a cyclotomic field F we define a canonical set of elements in K_3(F) and present a novel approach (based on a free differential calculus) to constructing them. Indeed, we are able to explicitly construct their images in H_{ 3}(GL(\mathbb{ C})) under the Hurewicz map. Applying our formula to these images yields a value V_1(F), which coincides with the Borel regulator R_1(F) when our set is a basis of K_3(F) modulo torsion. For F= \mathbb{ Q}(e^{ 2\pi i/3}) a computation of V_1(F) has been made based on our techniques.
Categories: math.KT math.NT