Help w/ Euclidean Rings and P.I.D.s
Hey, I have two questions I need some help with because I missed my recitation. Answers are fine, but I'd prefer it if someone could just tell me the procedure/give me a nudge in the right direction.
The first is: Suppose that in the Euclidean Algorithm for finding the gcd of two integers "a" and "b" we always choose "q" and "r" so a=bq+r and 0 = or < r < b. How many applications of the division algorithm are needed to find the gcd of F(n) and F(n+1) in where F is the fibonnaci sequence and F(0)=1. What is this gcd?
And the second: Show that if R is a Principal Ideal Domain and D is a multiplicatively closed subset of R, then (D^-1)R is a Principal Ideal Domain.
Thanks in advance.