On Thu, May 15, 2008 at 4:00 PM, John H Palmieri <[EMAIL PROTECTED]> wrote:
>
> The fact that i is not in CC doesn't bother me too much, since i is a
> "formal square root of -1":
>
> sage: i in CC
> False
>
> I can force it to be in CC by doing this:
>
> sage: CC(i)
> 1.00000000000000*I
>
> But then I think I should be able to do this:
>
> sage: GF(5)(i)
>
> Someone who knows more number theory than I do can come up with a
> better solution, but coercion code like this ought to work: replace
> the end ("raise TypeError...") of the _coerc_impl() method for
> Finite_Field_prime_modn (in sage.rings.finite_field_prime_modn.py)
> with this:
>
> from sage.functions.constants import I
> from sage.rings.arith import primitive_root
> p = self.__char
> if x == i and p % 4 == 1:
> return self(primitive_root(p))**((p-1)/4)
> raise TypeError, "no canonical coercion of x"
>
> But this isn't the right place to patch. When I do this,
> GF(5)._coerce_(i) works, but GF(5)(i) doesn't.
>
> Perhaps IntegerMod in the file integer_mod.pyx should be patched, but
> I don't want to mess around with Cython. Perhaps some other file
> should be patched. Can anyone help?
You should put the code in __call__. _coerce_ is only
for canonical coercions, which the above definitely isn't.
By the way, it would be (asymptotically) a lot faster to
choose a random element of GF(p) and raise it to the
power (p-1)/4 and check the resulting power is not +/-1.
Finding a primitive root requires factoring p-1, which is
potentially very slow.
-- William
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