the lower "k"-groups were discovered first, in the sense that adequate descriptions of these groups in terms of other algebraic structures were found. for example, if "f" is a field, then is isomorphic to the integers z and is closely related to the notion of vector space dimension. for a commutative ring "r", is related to the picard group of "r", and when "r" is the ring of integers in a number field, this generalizes the classical construction of the class group. the group "k"("r") is closely related to the group of units , and if "r" is a field, it is exactly the group of units. for a number field "f", "k"("f") is related to class field theory, the hilbert symbol, and the solvability of quadratic equations over completions. in contrast, finding the correct definition of the higher "k"-groups of rings was a difficult achievement of daniel quillen, and many of the basic facts about the higher "k"-groups of algebraic varieties were not known until the work of robert thomason.