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@nbruin ? I don't know about copying

I'm not sure what the copying behaviour should be for this structure. Reading the intro of the file, there is a very particular reason why the algebraic closure is not UniqueRepresentation. It is EqualityById, which looks like it's not a problem.
This explains why unpickling produces a new instance ... although

k=GF(5)
kbar=k.algebraic_closure()
GF(5).algebraic_closure() == kbar ##True

suggests that indirectly there IS a preferred copy: the one cached on the corresponding prime field.

The fact that multiple copies can exist would suggest that at least deep_copy should be able to produce a separate copy. Perhaps check with Peter Bruin @pjbruin ?

At the moment, only prime finite fields have algebraic closures. As far as I can see, prime finite fields are not formally UniqueRepresentation, but producing a finite field from GF(p) twice does produce identical results. So similarly, perhaps its algebraic closure is effectively also UniqueRepresentation, in which case we might want to fix pickle and then it wouldn't be controversial for copy to just return self.

More particularly:

sage: k=GF(5)
sage: loads(dumps(k)) is k
True
sage: loads(dumps(kbar)) is kbar ## this is more unequal ...
False
sage: loads(dumps(k)).algebraic_closure() is kbar ## than this
True

and I think the following is actually problematic

sage: kbar.prime_subfield().algebraic_closure() is kbar ##one would expect this
True
sage: kcopy = loads(dumps(kbar))
sage: kcopy.prime_subfield().algebraic_closure() is kcopy ## but that's broken on the copy
False