Hi Justin, You can see the problem from:
sage: O.ambient() Number Field in a0 with defining polynomial x^2 - 10 with a0 = a As you can see, the ambient field of O is not identical to K (which is the ambient field of OK). It is a field with an *embedding* into K, though, and it happens to be an isomorphism here. Mathematically, I think sage is right to have some reservation here. If we do sage: K.<a>=NumberField(x^4-2) sage: OK=K.ring_of_integers() sage: b=a^2 sage: O=ZZ[b] sage: O.ambient() Number Field in a0 with defining polynomial x^2 - 2 with a0 = a^2 I think you see the problem: the field of fractions of ZZ[b] does not need to be (and in the above example isn't) equal to the parent of b. In this case, O wouldn't have a finite index in OK. For programmatic consistency, we generally avoid programming in "shortcuts" based on specific values: while ZZ[a] happens to be an order in the parent K of a if a generates it over QQ, we'd generally *not* special case that to then create ZZ[a] as an order in K, but still create this separate field with generator a0. It's perhaps inconvenient and pedantic, but I would think this one may even still be on the instructive side for having to explain to a student. On Wednesday, 6 September 2023 at 17:21:12 UTC-7 Justin C. Walker wrote: > Hi, all, > > I think I understand what’s going wrong, but I don’t understand how to fix > the following problem: > > sage: K.<a>=NumberField(x^2-10) > sage: OK=K.maximal_order() > sage: O=ZZ[a] > sage: a in OK > True > sage: a in K > True > sage: a in O > True > sage: O.index_in(OK) > --------------------------------------------------------------------------- > ValueError Traceback (most recent call last) > ...Blooie > ValueError: other must have the same ambient number field as self. > > Have we just painted ourselves into a (figurative) corner? > > Pointers? Suggestions? > > Thanks for any help. > > Justin > > -- > Justin C. Walker, Curmudgeon at Large > Institute for the Absorption of Federal Funds > ----------- > While creating wives, God promised men > that good and obedient wives would be > found in all corners of the world. > Then He made the earth round. > -- > > > > > > > > > -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/sage-devel/91b7f3ea-fd89-446e-8a26-3e474414d32cn%40googlegroups.com.
