ABSTRACT
Over a Euclidean domain, in particular the ring F[x] of polynomials in one variable x over the field F, a modification of Gaussian elimination gives a similar result. Since the domain is Euclidean, it is also a PID, and this means that we can implement the Euclidean algorithm for gcds using elementary row operations. Thus during every iteration of row reduction, we use elementary row operations to replace the elements at and below the pivot by a single nonzero element at the pivot, which is the monic gcd of the original elements, together with zeros below the pivot. This analogue of the RCF is called the Hermite normal form (HNF). The same comments apply to column operations, and combining elementary row and column operations produces an algorithm for computing the Smith form of a matrix with entries in a Euclidean domain.
