Hi Bruno On 10 Feb., 12:26, Bruno Le Floch <blfla...@gmail.com> wrote: > True. But in the case of Q (and more generally in the case of the > quotient field of a (principal?) ring), we can be consistent with the > ring of integers, without any guess-work.
Sure. This could be one of the definitions I mentioned: lcm(a/b,c/d) = lcm(a,c)/gcd(b,d). But: Would it be a wise idea to have a totally different definition for fields and for quotient fields? Let me phrase it like this: There are different interpretations of the term "consistent". On the one hand, one could mean "consistency with respect to sub- structures": Let S be a sub-ring of a ring R; gcd_R is consistent with gcd_S <=> gcd_R(x,y)==gcd_S(x,y) for all x,y in S. As you have pointed out, there is a way to define the gcd of a quotient field consistent with the gcd of the underlying ring. On the other hand, one could mean "consistent as algebraic notion" (or perhaps "consistent with respect to categories"). Let me elaborate: One could argue that there is a partial map "gcd_*" that assigns to any object R of Rings() a function gcd_R that accepts two arguments (namely elements of R) returning elements of R. Now, it is possible to consider the same object R as object of different categories. For example, we have sage: QQ in QuotientFields() True sage: QQ in Fields() True sage: QQ in PrincipalIdealDomains() True So, it makes sense to call gcd_* "consistent as algebraic notion", if gcd_R for R object of some category C is the same as gcd_R for the same R considered as an object of a different category C'. What you propose would be "consistent with respect to subrings" ( gcd(QQ(m),QQ(n))==gcd(m,n) for m,n in ZZ), but it would be "inconsistent as algebraic notion", as gcd(a/b,c/d) would depend on whether we consider QQ as a quotient field or as a principal ideal domain. I strongly prefer to work with things that are consistent as algebraic notions. Best regards, Simon -- To post to this group, send an email to sage-devel@googlegroups.com To unsubscribe from this group, send an email to sage-devel+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-devel URL: http://www.sagemath.org
