In reviewing #2340, which adds some nice documentation for
rings/ring.pyx -- done by cswiercz -- I came across for the first time
the feature that gcd(a,b) works for rational a,b, returning a
perfectly sensible answer.

sage: gcd(1/2,1/3)
1/6

Are there any other fields for which someone has implemented a gcd function?

Although this is a nice function and I cann that it has its uses, I am
not so happy with its name....   I could just as easily make a case
for gcd(a,b)=1 for all rationals a,b (except (gcd(0,0)=0).

-- 
John Cremona

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