Subfields of finite fields

[Rob Beezer]

Is there a way to build subfields of finite fields that will behave as subfields?

For example, a finite field of order 3^6 will have proper subfields of order 3^1, 3^2, 3^3.  The first is not too interesting and can be recovered with the .prime_subfield() method.  I can easily build the set of elements for the other two, using the cyclic group structure of the nonzero elements under multiplication.  But these are just lists, with no structure.  For teaching purposes I'd like to get back a finite field that "checks out" as a subring or subfield.

I tried building quotient rings of the ring of polynomials over Z_3, without much luck, but I was not very careful about my choices for irreducible polynomials.  So maybe there is some way to make those choices and create two quotient rings that are finite fields with a subfield relationship.

Any suggestions or pointers would be appreciated.

Thanks,
Rob

[P Purkayastha]

According to sage/rings/finite_rings/finite_field_givaro.py it seems it is not implemented?

    def _coerce_map_from_(self, R):
        """
        Returns True if this finite field has a coercion map from R.
        ...
        """
        ...
               if R.degree() == 1:

                   return True

               elif self.degree() % R.degree() == 0:

                   # This is where we *would* do coercion from one nontrivial finite field to another...

                   # We use this error message for backward compatibility until #8335 is finished

                   raise TypeError, "unable to coerce from a finite field other than the prime subfield"

[David Loeffler]

On Sunday, 8 April 2012 08:43:03 UTC+1, P Purkayastha wrote:

According to sage/rings/finite_rings/finite_field_givaro.py it seems it is not implemented?

There have been at least two attempts to implement this; see #8335, and more recently, #11938. Sadly both of these are "needs work". It would be great if someone could finish either of these patches!

David

[Rob Beezer]

Thanks, David.  I'll watch those tickets.  -Rob

[Daniel Krenn]

Am 2012-04-08 10:16, schrieb David Loeffler:
> There have been at least two attempts to implement this; see #8335, and
> more recently, #11938. Sadly both of these are "needs work". It would be
> great if someone could finish either of these patches!

I've updated #11938 and marked it as need review.

Daniel