Abstract
This chapter describes the K2 of polynomial rings and of free algebras. The K2 functor of Milnor has been computed for certain fields. It turns out that some deep number theory is required in the process. By contrast, K2 of polynomial rings can be reduced to K2 of the ground field, using only a presentation of GLn for these polynomial rings. To show that R is a generalized Euclidean (GE) ring, one has to express every invertible matrix as a product of elementary matrices. It is found that to prove that R is universal for GEn, one has to show that every product of elementary matrices can be brought to a certain standard form, and the only product in this standard form equal to I is the empty product. It is assumed that R is a ring with a weak algorithm relative to a degree function. It is found that R is a GE-ring and if one can show it to be universal, this will certainly cover the case of free k-algebras.
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