Continuing in this abstract vein, I think I'd argue for letting GCD(x,y) evaluate to itself ("GCD(x,y)") in the event that the ideal generated by x and y is non-principal. Imagine a situation in which the ideal generated by x and y is principal, as is the ideal generated by x, y, and z, but the ideal generated by y and z is not. We'd like our computer algebra system to give the same output when evaluating GCD(GCD(x,y),z) or GCD(x,GCD(y,z)). On the other hand, I think it'd be too much to expect a GCD routine to have too many nice formal properties if we're going to subject it to such abuse. Come to think of it, if we're really going to use a GCD routine in rings other than Z, we need a general rule for breaking ties between elements of the ring and their associates. It's clear how to do this in Z[i] or Q[x,y], but unclear to me how to do this in general. Anyway, while there are things about Mathematica that I don't like, I respect its tendency to answer hard questions by repeating the question back to me (as if to say, "You tell me!"). Jim Propp On Thu, Nov 28, 2013 at 1:27 PM, Eugene Salamin <gene_salamin@yahoo.com>wrote:
Consider GCD ring theoretically. If a and b are elements of a ring R (which really should be an integral domain), GCD(a,b) is the smallest ideal in R containing a and b, and is the set of elements of the form ax + by as x and y range over R. An ideal containing 1 can only be the entire ring. If the ring is a field, then the only ideals are {0} and the entire field. GCD(a,b) depends on the containing ring R. If we take R to be the real numbers, then GCD(1,π) is R, which is also the ideal (a) for any nonzero a. So one can let GCD(1,π) be any nonzero real number.
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