Starting in the late 1960s, Collins and Brown & Traub invented
polynomial remainder sequences (PRS) in order to apply the Euclidean
algorithm to integer polynomials. Subresultants play a major role in
this theory. We compare the various notions of subresultants, give a
general and precise definition of PRS, and clean up some loose ends:
Lastly, we show how Kronecker had, already in the 1870s, discovered many of the fundamental properties of Euclid's algorithm for polynomials.