Implement integer_representation method for all finite fields

[slelievre]

In Sage 9.3.rc0, this works:

sage: p = 127
sage: K.<t> = GF(p^2, modulus=x^2 + 1)
sage: type(K)
<class
'sage.rings.finite_rings.finite_field_givaro.FiniteField_givaro_with_category'>
sage: a = 3*t + 2
sage: a.integer_representation()
383
sage: 3*p + 2
383

but this does not:

sage: p = 367
sage: K.<t> = GF(p^2, modulus=x^2 + 1)
sage: type(K)
<class
'sage.rings.finite_rings.finite_field_pari_ffelt.FiniteField_pari_ffelt_with_category'>
sage: a = 3*t + 2
sage: a.integer_representation()
Traceback (most recent call last)
...
AttributeError:
'sage.rings.finite_rings.element_pari_ffelt.FiniteFieldElement_pari_ffelt'
object has no attribute 'integer_representation'

Expected (3 * 367 + 2):

1103

Additionally, the method should return a Sage integer rather than a Python integer as the existing implementation for Givaro finite field elements currently does.

Suggested by revisiting this 2013 question by Ask Sage user "bonjour":

• ​Ask Sage question 9369: Finite field extension

Similar to:

• #31294: Implement fetch_int method for non-givaro finite fields

[gh-DaveWitteMorris]

This seems to be a duplicate of #8578 (except that the name of the method seems to have been int_repr back then), so I think one of the two should be closed.

There is a bit of discussion of this issue in ​this sage-devel thread.

[slelievre]

Replying to gh-DaveWitteMorris:

This seems to be a duplicate of #8578 (except that the name of the method seems to have been int_repr back then), so I think one of the two should be closed.

Thanks for spotting that. I agree one of the tickets should be closed as duplicate. I don't mind which. Maybe it's less trouble to close the older one given the method changed name in the meantime.

[mkoeppe]

Sage development has entered the release candidate phase for 9.3. Setting a new milestone for this ticket based on a cursory review.

[lorenz]

Sorry, I wasn't aware of this ticket and opened yet another duplicate at #33941. (This time with a patch!)