On 2 May 2009, at 04:05, Steve wrote:
Why is gcd 0 0 undefined?
In math, one may define gcd(x, y) as a generator of the ideal generated by x and y in the ring of integers Z. The gcd(x, y) then always exists as the ring Z is a PID (principal ideal domain), i.e., all ideals can be generated by a single element, which can be proven using Euclid's algorithm, also useful for computing the gcd in Z.
Anyway, the ideal generated by 0 and 0 is the zero ideal 0, which also is generated by the single generator 0. So gcd(0, 0) = 0 by this definition.
In Z, one may take the gcd >= 0, but that may not work in every PID. If k is a field, then the polynomial ring k[x] is a PID, but not the ring k[x_1, ..., x_n]. So that leads to Buchberger's Groebner Basis Algorithm.
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