>> Let me phrase it like this: There are different interpretations of the >> term "consistent".
@Simon: You are right to distinguish the two kinds of consistencies. And I can understand that sometimes it is preferable to have the algebraic consistency. I tend to care about elements of the objects more than the objects of the category (i.e. individual rational numbers rather than the field/PID/quotient field QQ), and thus I tend towards subring consistency. > You could have both consistencies. That depends on how you define gcd > and lcm: > > - Quotient fields as described by Bruno. > - Fields: zero if both elements are zero. A non-zero element > otherwise (most fields would choose 1 here). > - PID: a generator of the corresponding ideal. I don't see how this brings in both consistencies. Algebraic consistency requires gcd and lcm on QQ to have different outputs depending on whether QQ is seen a Field, a PID, a Quotient Field... Is there a clear way for the user to indicate "which QQ" he wants? Or we could have (I don't really know how this is done ;-) ) lcm(10/21, 14/15, type="PID") = 1 lcm(10/21, 14/15, type="Field") = 1 lcm(10/21, 15/14, type="quotient-of-ZZ") = 30/7 I doubt that the "field" version is useful at all: the lcm is basically always 1 (except when one of the arguments is zero). lcm and gcd should only be defined for PIDs, where they are interesting (or for factorization rings? I can't remember my undergrad). > http://groups.google.com/group/sage-devel/browse_thread/thread/12524b18d2325633/7b8af907c3c45c8b?lnk=gst&q=gcd+and+lcm+for+field+elements#7b8af907c3c45c8b Interesting read, thanks. -- 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
