Given an irreducible polynomial P from Q[x] I want to find the primitive
element A of the extension field defined by it.
Moreover I want to know how to write the roots of P in terms of A(per
example, the coefficients of the polynomial of powers of A).
Finally I want to find the dimension of the vectors space of the extension
field of the roots of P.
For example, if the roots are sqrt(2) and -sqrt(2) I can write one in terms
of the other with rational coefficients(they are rational dependent). So
they have dimension 1, in some sense.
One such way of starting this is by making the following:
P = some irreducible polynomial on variable x
K.<a> = NumberField(P)
f = (a).coordinates_in_terms_of_powers()
rts = P.roots(ring=QQbar)#need the QQbar as some without this some roots
are not returned
for r in rts:
coeff = f(r[0])
#do something with it
So my idea was using this f to get the coefficients of the polynomial in A
of the roots of P. But when I do the above I get the following error:
File "sage/rings/number_field/number_field_element.pyx", line 5341, in
sage.rings.number_field.number_field_element.CoordinateFunction.__call__
(build/cythonized/sage/rings/number_field/number_field_element.cpp:43366)
raise TypeError("Cannot coerce element into this number field") TypeError:
Cannot coerce element into this number field
As I saw r[0]'s parent is a Algebraic Field. There is any way of casting or
coercing this field into the number field in order to get the coefficients?
Because when I write a "number" M in terms of a and call f(M) it works as
expected. If not, is there any other way of getting this coefficients? Any
work around from number field that gives me these coefficients in terms of
the primitive element from the polynomial that generates it? Or is there
another approach to this using vectors spaces or modules? Because in the
end I'll also need the dimension as said before.
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