Hm, interesting. The current code for ZZ.__getitem__ (i.e., ZZ[a]) does
this:
if isinstance(x, NumberFieldElement_base):
K, _ = parent(x).subfield(x)
return K.order(K.gen())
However, this functionality is also available in K.order:
sage: K.order(a,allow_subfield=True)
Order in Number Field in a with defining polynomial x^4 - 2
sage: K.order(a^2,allow_subfield=True)
Order in Number Field in beta with defining polynomial x^2 - 2*x - 1 with
beta = -a^2 + 1
so the code could actually just be:
if isinstance(x, NumberFieldElement_base):
return K.order(x,allow_subfield=True)
and then Justin's problem would be solved. It's more economical in that no
(isomorphic) copy of the field is created. It looks like it does cause
different problems, though: it looks like K.order wants a standardised
basis, so:
sage: K.order(a+1).gens()
(1, a, a^2, a^3)
sage: K.order(a^2+a+1).gens()
(1, 7*a^3 + a, 2*a^3 + a^2, 9*a^3)
So K.order(a) does not need to return an order with a power basis on a.
Perhaps that's the deeper reason why ZZ[a] does its trickery: certainly one
would expect an order with power basis on a from that construction.
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