On 4 May 2015 at 15:22, Evrim Ulu <[email protected]> wrote:
>
> Here it is:
>
> F16.extension(modulus=x^7+x+1)
To quote from the documentation of the extension() method used here:
"Extensions of non-prime finite fields by polynomials are not yet
supported: we fall back to generic code:"
follwed by an example. In your case you get
Univariate Quotient Polynomial Ring in x over Finite Field in g of
size 2^4 with modulus x^7 + x + 1
which does at least know that it is a field, but its type is not
FiniteField but PolynomialQuatientRing .
There has been recent work to improve handling of relative extensions
of finite fields, and perhaps someone who has been involved with that
will comment further.
John Cremona
>
>
> On Monday, May 4, 2015 at 5:02:52 PM UTC+3, Evrim Ulu wrote:
>>
>> Hello,
>>
>> I'm having trouble extending a finite field. Any help would be
>> appreciated.
>>
>> F16 = GF(16, 'g')
>> F16_x.<x> = PolynomialRing(F16, 'x')
>> HH = GF(F16^7, modulus=x^7 + x + 1, name='h')
>>
>> I basically try to extend 2^4 to 2^4*7 with a degree 7 irreducible.
>> I get the following.
>>
>> best,
>> evrim.
>>
>>
>> sage: HH = FiniteField(F16^7, modulus=x^7 + x + g^15, name='h')
>>
>> ---------------------------------------------------------------------------
>> TypeError Traceback (most recent call
>> last)
>> /usr/lib/sagemath/local/lib/python2.7/site-packages/sage/all_cmdline.pyc
>> in <module>()
>> ----> 1 HH = FiniteField(F16**Integer(7), modulus=x**Integer(7) + x +
>> g**Integer(15), name='h')
>>
>> /usr/lib/sagemath/src/sage/structure/factory.pyx in
>> sage.structure.factory.UniqueFactory.__call__
>> (build/cythonized/sage/structure/factory.c:1207)()
>> 362 False
>> 363 """
>> --> 364 key, kwds = self.create_key_and_extra_args(*args, **kwds)
>> 365 version = self.get_version(sage_version)
>> 366 return self.get_object(version, key, kwds)
>>
>>
>> /usr/lib/sagemath/local/lib/python2.7/site-packages/sage/rings/finite_rings/constructor.pyc
>> in create_key_and_extra_args(self, order, name, modulus, names, impl, proof,
>> check_irreducible, **kwds)
>> 426 proof = arithmetic()
>> 427 with WithProof('arithmetic', proof):
>> --> 428 order = Integer(order)
>> 429 if order <= 1:
>> 430 raise ValueError("the order of a finite field must
>> be at least 2")
>>
>> /usr/lib/sagemath/src/sage/rings/integer.pyx in
>> sage.rings.integer.Integer.__init__
>> (build/cythonized/sage/rings/integer.c:6020)()
>> 688 return
>> 689
>> --> 690 raise TypeError, "unable to coerce %s to an
>> integer" % type(x)
>> 691
>> 692 def __reduce__(self):
>>
>> TypeError: unable to coerce <class
>> 'sage.modules.free_module.FreeModule_ambient_field_with_category'> to an
>> integer
>>
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