Math 513. Class Worksheet on Euclidean Algorithm.
Professor Karen E. Smith
Wednesday January 27, 2012
Let Rbe a Euclidean domain.
1. For non-zero elements f0, f1with σ(f0)≥σ(f1), write
f0=f1q1+f2,
where σ(f2)< σ(f1). Show that
(f0, f1)=(f1, f2)
as ideals in R.
2. Use this to find the greatest common divisor of 2210 and 78 (in Z).
3. Describe an algorithm to find the gcd of any two elements in a Euclidean domain. (This
technique is due to Euclid, though of course, he was only interested in the integers.)