"k"-theory was invented in the late 1950s by alexander grothendieck in his study of intersection theory on algebraic varieties. in the modern language, grothendieck defined only "k", the zeroth "k"-group, but even this single group has plenty of applications, such as the grothendieck–riemann–roch theorem. intersection theory is still a motivating force in the development of (higher) algebraic "k"-theory through its links with motivic cohomology and specifically chow groups. the subject also includes classical number-theoretic topics like quadratic reciprocity and embeddings of number fields into the real numbers and complex numbers, as well as more modern concerns like the construction of higher regulators and special values of l-functions.