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
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