Re: [sage-devel] Units in Orders
On 13/08/2012 06:50, David Roe wrote: Thanks for the pointer to that ticket, which explains the change in the the is_unit() behavior. Why should the inverse of four succeed when the result is not in K? sage: four^-1 in K False The order K is analogous to the ring of integers inside QQ. So even though the inverse of four exists in the number field, it's not in the order and thus four is not a unit. See http://en.wikipedia.org/wiki/Order_%28ring_theory%29 Exactly. This is completely analogous to the following: sage: 4.parent() is ZZ True sage: 4.is_unit() False sage: 4^-1 1/4 sage: 4^-1 in ZZ False sage: (4^-1).parent() is QQ True In your example, ZZ is replaced with the order K, and QQ is replaced by the field of fractions of K, which is the number field QQ(sqrt(-3)). On #12242 (a follow-on to the ticket above), David Loeffler argues that the following is the wrong behavior, and that the last command should raise an error. sage: K.a = NumberField(x^2 - x - 1); OK = K.ring_of_integers() sage: OK(12).divides(OK(13)) True Here again OK is a generalization of ZZ. In K, 12 does divide 13 and the quotient is 13/12, but since 13/12 is not in OK, the answer should be False as David says. sage: OK(12) // OK(13) 12/13 Since OK is a generalization of ZZ, we look at what // does for ZZ. From the reference manual, http://www.sagemath.org/doc/tutorial/tour_assignment.html : sage:10//4# for integer arguments, // returns the integer quotient 2 So // gives the integral part of the quotient, forgetting the remainder. Just like / does for python ints. Therefore, sage: 12//13 0 sage: (12//13).parent() is ZZ True So whatever OK(12) // OK(13) returns, it has to be an element of OK, hence (as David L. says: not 12/13). Moreover, we must always have b * (a // b) + (a % b) == a for this quotient and remainder. The easiest fix is have it raise an Error, which is better than anything that is not in OK. Again, it's a question of where the arithmetic is taking place. The issue is that 12/13, while an element of K, is not in OK. David -- -- 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
Re: [sage-devel] Units in Orders
On Saturday, August 11, 2012 1:22:05 PM UTC-7, Marco Streng wrote: These outputs look fine to me. See also http://trac.sagemath.org/sage_trac/ticket/11673 Dear Marco, Thanks for the pointer to that ticket, which explains the change in the the is_unit() behavior. Why should the inverse of four succeed when the result is not in K? sage: four^-1 in K False On #12242 (a follow-on to the ticket above), David Loeffler argues that the following is the wrong behavior, and that the last command should raise an error. sage: K.a = NumberField(x^2 - x - 1); OK = K.ring_of_integers() sage: OK(12).divides(OK(13)) True sage: OK(12) // OK(13) 12/13 Thanks, Rob -- -- 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
Re: [sage-devel] Units in Orders
Thanks for the pointer to that ticket, which explains the change in the the is_unit() behavior. Why should the inverse of four succeed when the result is not in K? sage: four^-1 in K False The order K is analogous to the ring of integers inside QQ. So even though the inverse of four exists in the number field, it's not in the order and thus four is not a unit. See http://en.wikipedia.org/wiki/Order_%28ring_theory%29 On #12242 (a follow-on to the ticket above), David Loeffler argues that the following is the wrong behavior, and that the last command should raise an error. sage: K.a = NumberField(x^2 - x - 1); OK = K.ring_of_integers() sage: OK(12).divides(OK(13)) True sage: OK(12) // OK(13) 12/13 Again, it's a question of where the arithmetic is taking place. The issue is that 12/13, while an element of K, is not in OK. David -- -- 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
[sage-devel] Units in Orders
Is this a bug? First example is from Judson's abstract algebra text. Note that .is_unit() returned True in sage 4.8. sage: sage: K.x,y = ZZ[sqrt(-3)]; K Order in Number Field in a with defining polynomial x^2 + 3 sage: four = K(4) sage: four.is_unit() False sage: four^-1 1/4 Second example: all but the last input is basically from the documentation of .is_unit(). sage: K.a = NumberField(x^2 - x - 1) sage: OK = K.ring_of_integers() sage: u = OK(13) sage: u.is_unit() False sage: u^-1 1/13 Thanks, Rob -- -- 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
Re: [sage-devel] Units in Orders
These outputs look fine to me. See also http://trac.sagemath.org/sage_trac/ticket/11673 2012/8/11 Rob Beezer goo...@beezer.cotse.net: Is this a bug? First example is from Judson's abstract algebra text. Note that .is_unit() returned True in sage 4.8. sage: sage: K.x,y = ZZ[sqrt(-3)]; K Order in Number Field in a with defining polynomial x^2 + 3 sage: four = K(4) sage: four.is_unit() False sage: four^-1 1/4 Second example: all but the last input is basically from the documentation of .is_unit(). sage: K.a = NumberField(x^2 - x - 1) sage: OK = K.ring_of_integers() sage: u = OK(13) sage: u.is_unit() False sage: u^-1 1/13 Thanks, Rob -- -- 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 -- -- 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
