Raising integers to finite-field elements coerces the base into the field

[yyyyx4]

In sage 9.3, raising an integer to a finite-field element in some cases coerces the base into the finite field. I think the correct behaviour should be to throw an error, or attempt to interpret the exponent as an integer (which is mathematically ill-defined but perhaps convenient).

Example:

sage: x = 123**GF(49)(5); x, x.parent()
(2, Finite Field in z2 of size 7^2)

Interestingly, this does not happen with prime finite fields, which inherit from integers modulo n:

sage: x = 123**GF(47)(5); x, x.parent()
(28153056843, Integer Ring)

Component: coercion

Keywords: finite fields, exponentiation

Issue created by migration from https://trac.sagemath.org/ticket/32287

[Shay2Shay]

I was looking into the issue and found some of the past issue related to this
#24247 : It allowed coercion in power operation.
#15709 : It mentioned some inconvenience caused due to coercion.

I believe this issue can be fixed by making some changes here.

Shall we add support for FiniteField_givaro too? (and others FiniteField_ntl_gf2e or FiniteField_pari_ffelt)

Additional Info:

sage: x = 123**GF(64, impl='givaro')(5); x, x.parent()
(1, Finite Field in z6 of size 2^6)
sage: x = 123**GF(64, impl='pari_ffelt')(5); x, x.parent()
(1, Finite Field in z6 of size 2^6)
sage: 
sage: x = 123**GF(64, impl='ntl')(5); x, x.parent()
...
...
...
TypeError: unsupported operand parent(s) for ^: 'Finite Field in z6 of size 2^6' and 'Finite Field in z6 of size 2^6'

What should be right course of action here ?

• Maybe we can remove feature
• Maybe we can provide support for givaro and others
• Maybe we can create a backend variable to handle such cases (similar to with proof which is used for primality testing)

[Shay2Shay]

I choose to dis-allow Finite Field with type p^n {n>1} to be used as argument for powering. (#38754)
Ex: 2 ** ( GF(2^6)(5) ) = Error

But this does not block earlier ModN implementation
Ex: 2 ** ( GF(29)(5) ) = No Error

I felt #38403 is also a similar issue