On Sunday, April 8, 2012 4:57:11 AM UTC+8, Rob Beezer wrote:
>
> 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
>
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"
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