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})"
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