Integral Domains, Euclidean Domains

Euclidean Domains

A euclidean domain is a ring with a metric d() that maps the nonzero elements of the ring into the positive integers, and satisfies d(a*b) ≥ d(a), and for any nonzero a and b there is c and r, such that a = c*b+r, and r = 0 or d(r) < d(b). By convention, we write d(0) = 0. Verify the following about d().

You can see how we're going to run the gcd algorithm with the remainders getting smaller and smaller, at least according to d(). This will allow us to prove unique factorization, just like we did for the integers. But we'll get to that later.