Error in operations of finite field

[Samanta]

I did not understand why the error has occurred. For the first time, it returned output but for the second time when I run the same code, I found an error "ValueError: the degree of the modulus does not equal the degree of the field" which is also not true as my irreducible polynomial has degree 6.
I am attaching my code and output for two consecutive times.

sage: K.<a>=GF(2^6, modulus=x^6 + x^4 + x^3 + x + 1)

....: x=a^9
....: y=a^21
....: z=1/x
....: v=vector([1,y,y^2,y^3,y^4,y^5,y^6])
....: S=matrix([[1,1,1,1,1,1,1],[1,z,z^2,z^3,z^4,z^5,z^6],[1,(z^2),(z^2)^2,(z^2)^3,(z^2)^4,(z^2)^5,(z^2)^6],[1,(z^3),(z^3)^2,(z^3)^3,(z^3)^4,(z^
....: 3)^5,(z^3)^6],[1,(z^4),(z^4)^2,(z^4)^3,(z^4)^4,(z^4)^5,(z^4)^6],[1,(z^5),(z^5)^2,(z^5)^3,(z^5)^4,(z^5)^5,(z^5)^6],[1,(z^6),(z^6)^2,(z^6)^3
....: ,(z^6)^4,(z^6)^5,(z^6)^6]])
....: w=S*v
....: w
....:

(1, a^3 + a + 1, a^5 + a^4 + a^3 + a^2 + 1, a^4 + a^3 + a^2, a^5 + a^4 + a^3, a^3 + a^2 + 1, a^4 + a^3 + a^2 + a + 1)

sage: K.<a>=GF(2^6, modulus=x^6 + x^4 + x^3 + x + 1)
....: x=a^9
....: y=a^21
....: z=1/x
....: v=vector([1,y,y^2,y^3,y^4,y^5,y^6])
....: S=matrix([[1,1,1,1,1,1,1],[1,z,z^2,z^3,z^4,z^5,z^6],[1,(z^2),(z^2)^2,(z^2)^3,(z^2)^4,(z^2)^5,(z^2)^6],[1,(z^3),(z^3)^2,(z^3)^3,(z^3)^4,(z^
....: 3)^5,(z^3)^6],[1,(z^4),(z^4)^2,(z^4)^3,(z^4)^4,(z^4)^5,(z^4)^6],[1,(z^5),(z^5)^2,(z^5)^3,(z^5)^4,(z^5)^5,(z^5)^6],[1,(z^6),(z^6)^2,(z^6)^3
....: ,(z^6)^4,(z^6)^5,(z^6)^6]])
....: w=S*v
....: w
....:
---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
<ipython-input-3-37bf7e6f8848> in <module>()
----> 1 K = GF(Integer(2)**Integer(6), modulus=x**Integer(6) + x**Integer(4) + x**Integer(3) + x + Integer(1), names=('a',)); (a,) = K._first_ngens(1)
      2 x=a**Integer(9)
      3 y=a**Integer(21)
      4 z=Integer(1)/x
      5 v=vector([Integer(1),y,y**Integer(2),y**Integer(3),y**Integer(4),y**Integer(5),y**Integer(6)])

/home/susanta/Desktop/SageMath/local/lib/python2.7/site-packages/sage/structure/factory.pyx in sage.structure.factory.UniqueFactory.__call__ (build/cythonized/sage/structure/factory.c:2162)()
    366             False
    367         """
--> 368         key, kwds = self.create_key_and_extra_args(*args, **kwds)
    369         version = self.get_version(sage_version)
    370         return self.get_object(version, key, kwds)

/home/susanta/Desktop/SageMath/local/lib/python2.7/site-packages/sage/rings/finite_rings/finite_field_constructor.pyc in create_key_and_extra_args(self, order, name, modulus, names, impl, proof, check_irreducible, prefix, repr, elem_cache, **kwds)
    583
    584                     if modulus.degree() != n:
--> 585                         raise ValueError("the degree of the modulus does not equal the degree of the field")
    586                     if check_irreducible and not modulus.is_irreducible():
    587                         raise ValueError("finite field modulus must be irreducible but it is not")

ValueError: the degree of the modulus does not equal the degree of the field

[Dima Pasechnik]

yes, I can confirm this is still giving an error in the latest Sage
beta. (9.0.beta6).
Note that actually it's two identical inputs, the 2nd executed after
the 1st gives the error.

I'll open a ticket.

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[Dima Pasechnik]

This is now https://trac.sagemath.org/ticket/28786
Note that it only happens if modulus is explicitly given.

[Dima Pasechnik]

This is a variable name clash. Somehow, using 'x' for modulus creates a problem.
If I instead do

T.<xx>=GF(2)[]
K.<a>=GF(2^6, modulus=xx^6 + xx^4 + xx^3 + xx + 1)
...

then everything works as expected.

[SUSANTA SAMANTA]

Thanks a lot.

T.<xx>=GF(2)[]
K.<a>=GF(2^6, modulus=xx^6 + xx^4 + xx^3 + xx + 1)

is working perfectly.

Also adding T.<x>=GF(2)[] before K.<a>=GF(2^6, modulus=x^6 + x^4 + x^3 + x + 1) works.

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[David Roe]

The issue is that you redefined x after defining K.  I wrote more on the ticket.

David

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[SUSANTA SAMANTA]

Thanks David for point out the issue. But when I get the error for the code, this error happens for the next code as well as like when I run

K.<a>=GF(2^6, modulus=x^6 + x^4 + x^3 + x + 1)

K

But the error goes away if I run (told by Dima )

T.<x>=GF(2)[]

K.<a>=GF(2^6, modulus=x^6 + x^4 + x^3 + x + 1)

K

So I think there may be a bug.

I am attaching the code and their corresponding output or error

....: K

....:
---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)

<ipython-input-4-7687ba85abff> in <module>()

----> 1 K = GF(Integer(2)**Integer(6), modulus=x**Integer(6) + x**Integer(4) + x**Integer(3) + x + Integer(1), names=('a',)); (a,) = K._first_ngens(1)

      2 K

/home/susanta/Desktop/SageMath/local/lib/python2.7/site-packages/sage/structure/factory.pyx in sage.structure.factory.UniqueFactory.__call__ (build/cythonized/sage/structure/factory.c:2162)()
    366             False
    367         """
--> 368         key, kwds = self.create_key_and_extra_args(*args, **kwds)
    369         version = self.get_version(sage_version)
    370         return self.get_object(version, key, kwds)

/home/susanta/Desktop/SageMath/local/lib/python2.7/site-packages/sage/rings/finite_rings/finite_field_constructor.pyc in create_key_and_extra_args(self, order, name, modulus, names, impl, proof, check_irreducible, prefix, repr, elem_cache, **kwds)
    583
    584                     if modulus.degree() != n:
--> 585                         raise ValueError("the degree of the modulus does not equal the degree of the field")
    586                     if check_irreducible and not modulus.is_irreducible():
    587                         raise ValueError("finite field modulus must be irreducible but it is not")

ValueError: the degree of the modulus does not equal the degree of the field

sage: T.<x>=GF(2)[]
....: K.<a>=GF(2^6, modulus=x^6 + x^4 + x^3 + x + 1)
....: K
....:
Finite Field in a of size 2^6

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[Dima Pasechnik]

On Fri, Nov 22, 2019 at 6:16 PM SUSANTA SAMANTA
<susantas...@gmail.com> wrote:
>
> Thanks David for point out the issue. But when I get the error for the code, this error happens for the next code as well as like when I run
> K.<a>=GF(2^6, modulus=x^6 + x^4 + x^3 + x + 1)
> K
> But the error goes away if I run (told by Dima )
> T.<x>=GF(2)[]
> K.<a>=GF(2^6, modulus=x^6 + x^4 + x^3 + x + 1)
> K
> So I think there may be a bug.

no, it's because you are using x for two different purposes. You can
get the same error just by doing

sage: T.<x>=GF(2)[]
....: K.<a>=GF(2^6, modulus=x^6 + x^4 + x^3 + x + 1)

....: x = a^9

....: K.<a>=GF(2^6, modulus=x^6 + x^4 + x^3 + x + 1)

it's not really a bug in Sage, it's a bug in your code.

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