#14990: Implement algebraic closures of finite fields
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Reporter: pbruin | Owner:
Type: enhancement | Status: needs_work
Priority: major | Milestone: sage-5.13
Component: algebra | Resolution:
Keywords: finite field | Merged in:
algebraic closure | Reviewers:
Authors: Peter Bruin | Work issues:
Report Upstream: N/A | Commit:
Branch: u/pbruin/14990 | 4265b4fe176d32b3bc7cbc616f8f0964e273ecdb
Dependencies: #14958, #13214 | Stopgaps:
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Comment (by pbruin):
Just to get an idea of what would be good to do before setting this to
"needs-review", here is the status of the items from Vincent's list
(comment:9):
> - .some_elements() returns a very stupid list
This can be made less stupid by implementing `_an_element_()` to return
`self.gen(2)`.
> - .gens() is not happy. I guess we can return a `Family`. The problem
is that in the specification it is written that the method should return a
tuple.
I think returning a `Family` is acceptable given that there is no finite
set of generators.
> - many methods can be implemented in two lines as
> {{{
> def my_method(self):
> return self._value.my_method()
> }}}
> this concerns `minimal_polynomial`, `trace`, `norm`,
`multiplicative_order`, ...
I agree for `minimal_polynomial` and `multiplicative_order`; the latter
was implemented by Vincent.
The trace and norm, on the other hand, are not well-defined since the
definition requires a ''finite'' extension, for which there is no canical
choice.
It remains to implement `minimal_polynomial`, which can of course be done
in two lines as above.
> - in a similar way (calling apropriate method of _value and applying a
morphism) it is straightforward to implement `frobenius`, `pth_power`,
`pth_root`, ...
We now have `pth_power` and `pth_root`, again thanks to Vincent, and the
field itself has `frobenius`. I don't think an additional `frobenius` on
elements would be useful.
--
Ticket URL: <http://trac.sagemath.org/ticket/14990#comment:49>
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