Indeed, this appears to have never worked, but the issue is not to do with choices of embeddings. In fact cyclotomic fields are implemented with a preferred choice of embeddings: the map from CyclotomicField(m) to CyclotomicField(m*n) always takes zeta_m to zeta_n^m.
There is some interesting archaeology to be done in number_field.py, where this problem seems to originate; the function _coerce_from_other_cyclotomic_field is the offender. There is some commented-out code that tried to fix this; the author indicates that the fix failed because it broke doctests for Eisenstein submodules. I didn't try to figure out who wrote this comment, or how long ago, but it appears to be not recent. Anyway, since there is a working method as Vincent points out, it should be a relatively easy fix to patch this in. Kiran On Monday, April 20, 2020 at 1:23:22 AM UTC-7, vdelecroix wrote: > > John, > > I don't think it ever worked. However it is a desirable feature > to have. As I mentioned on the ticket, there is already code for > this (which works for general number fields) > > sage: K12.<z12> = CyclotomicField(12) > sage: K3.<z3> = CyclotomicField(3) > sage: f = K3.hom([z12^2 - 1]) > sage: f.preimage(-3*z12^2 + 3) > -3*z3 > > This is basically what the code for K3(-3*z12^2 + 3) should be > doing. > > Best > Vincent > > Le 20/04/2020 à 09:45, John Cremona a écrit : > > Kiran, > > > > Did this ever work? > > > > When I have needed to work in several nested cyclotomic fields I have > > done this using the Lord of the Rings strategy "One ring to contain > > them all": define the largest one first and then define the subfields > > with explicit embeddings. Taken from code at > > https://arxiv.org/abs/1306.6818 I have The following lines which first > > define Q(zeta_1092) and then various subfields. All that worked. > > However I tried changing your second line to > > K1.<z1> = CyclotomicField(3, embedding=z^4) > > and it did not help. > > > > John > > > > Q1092.<zeta1092> = CyclotomicField(1092) > > conj1092 = Q1092.hom([zeta1092^(-1)]) # complex conjugation on Q1092 > > # > > Q13.<zeta13> = CyclotomicField(13, embedding=zeta1092^84) > > eQ13Q1092 = Q13.hom([Q1092(zeta13)]) # embedding Q13 into Q1092 > > Q13sigma = Q13.hom([zeta13^4]) # auto of order 3 of Q13 > > Q13aut = Q13.hom([zeta13^2]) # auto of order 6 of Q13 > > # > > Q91.<zeta91> = CyclotomicField(91,embedding=zeta1092^12) > > eQ91Q1092 = Q91.hom([zeta1092^12]) # embedding Q91 into Q1092 > > eQ13Q91 = Q13.hom([zeta91^7]) # embedding Q13 into Q91 > > Q91aut = Q91.hom([zeta91^53]) # auto of order 3 of Q91 fixing > Q13 > > # > > Q84.<zeta84> = CyclotomicField(84, embedding=zeta1092^13) > > conj84 = Q84.hom([zeta84^(-1)]) # complex conjugation on Q84 > > Q12.<zeta12> = CyclotomicField(12, embedding=zeta84^7) > > Q7.<zeta7> = CyclotomicField(7, embedding=zeta84^12) > > Q1092._latex_ = lambda: "\mathbb{Q}(\zeta_{1092})" > > Q13._latex_ = lambda: "\mathbb{Q}(\zeta_{13})" > > Q91._latex_ = lambda: "\mathbb{Q}(\zeta_{91})" > > Q84._latex_ = lambda: "\mathbb{Q}(\zeta_{84})" > > Q12._latex_ = lambda: "\mathbb{Q}(\zeta_{12})" > > Q7._latex_ = lambda: "\mathbb{Q}(\zeta_{7})" > > zeta7p=zeta7+1/zeta7 > > Q7p.<zeta7p>=NumberField(zeta7p.minpoly(), embedding=zeta7+1/zeta7) > > # > > zeta13pp = zeta13+zeta13^5+zeta13^8+zeta13^12 # generates cubic subfield > > Q13pp.<zeta13pp> = NumberField(zeta13pp.minpoly(), embedding=zeta13pp) > > eQ13ppQ13 = Q13pp.hom([Q13(zeta13pp)]) > > # > > zeta13c = zeta13+zeta13^3+zeta13^9 # generates quartic subfield > > Q13c.<zeta13c> = NumberField(zeta13c.minpoly(), embedding=zeta13c) > > eQ13cQ13 = Q13c.hom([Q13(zeta13c)]) > > Q7p._latex_ = lambda: "\mathbb{Q}(\zeta_{7}^{+})" > > Q13pp._latex_ = lambda: "\mathbb{Q}(\zeta_{13}^{++})" > > Q13c._latex_ = lambda: "\mathbb{Q}(\zeta_{13}^{c})" > > > > On Sat, 18 Apr 2020 at 23:30, Kiran Kedlaya <[email protected] > <javascript:>> wrote: > >> > >> I've already created ticket #29511 for this, but to advertise this more > broadly: there seems to be an issue with the canonical embedding of one > cyclotomic field into another, where elements that come from the subfield > cannot always be coerced back. > >> {{{ > >> > >> sage: K.<z> = CyclotomicField(12) > >> sage: K1.<z1> = CyclotomicField(3) > >> sage: K(z1) in K1 > >> True > >> sage: K1(K(z1)) > >> z1 > >> > >> sage: K(2) in K1 ## This is the source of the trouble > >> > >> False > >> > >> sage: K1(K(2)) ## Error! > >> > >> ... > >> > >> }}} > >> > >> > >> I am up for trying to fix this myself, but I probably need some help > from someone who understands coercion in Sage better than I do. > >> > >> > >> Kiran > >> > >> > >> -- > >> You received this message because you are subscribed to the Google > Groups "sage-nt" group. > >> To unsubscribe from this group and stop receiving emails from it, send > an email to [email protected] <javascript:>. > >> To view this discussion on the web visit > https://groups.google.com/d/msgid/sage-nt/d4708760-19b2-4a5c-a0b4-756eb592b08c%40googlegroups.com. > > > > > -- You received this message because you are subscribed to the Google Groups "sage-nt" group. 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