If you want the order generated by a, you can do sage: K.<a>=NumberField(x^2-10) sage: OK=K.maximal_order() sage: O = K.order(a) sage: O.index_in(OK) 1
Note that this correctly fails in Nils' example: sage: K.order(a^2) ... ValueError: the rank of the span of gens is wrong David On Thu, Sep 7, 2023 at 11:54 AM Nils Bruin <nbr...@sfu.ca> wrote: > 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 > <https://groups.google.com/d/msgid/sage-devel/91b7f3ea-fd89-446e-8a26-3e474414d32cn%40googlegroups.com?utm_medium=email&utm_source=footer> > . > -- 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/CAChs6_m%2BDdvtLUq5ouCEkNK6qo31upfkHqCLF_0R0DwTgmU1Nw%40mail.gmail.com.
