On 4 May 2015 at 13:48, Jeroen Demeyer <jdeme...@cage.ugent.be> wrote:
> On 2015-05-04 14:39, Bruno Grenet wrote:
>>
>> Dear all,
>>
>> 1. In number fields, some elements are considered as prime, which is
>> not mathematically correct:
>> |
>> sage:S.<x>=NumberField(x^2+5)
>> sage:S(11).is_prime()
>> True
>
>
> This is really due to S.ideal(11) returning a *fractional* ideal. I think
> it's difficult to change that now.
>
>> 2. When one defines a number field as above, one cannot define a new
>> number field anymore:
>> |
>> sage:S.<x>=NumberField(x^2+5)
>> sage:R.<y>=NumberField(x^2+7)
>> Traceback(most recent call last):
>> ...
>> ValueError:variable names must be alphanumeric,but one is'Rational
>> Field'which isnot.
>
Another comment: in your first number field definition you are tacitly
using the fact that on startup, Sage assigns to the python identifier
'x' the value of the symbolic variable x. That is why you can say
(immediately after startup):
K = NumberField(x^2+5)
but if you were to try
K = NumberField(y^2+5)
it would raise an error since y is not defined. Then, you are reusing
that identifier 'x' to be the generator of your first field, since the
line
S.<x>=NumberField(x^2+5)
is expanded by the preprocessor:
sage: preparse('S.<x>=NumberField(x^2+5)')
"S = NumberField(x**Integer(2)+Integer(5), names=('x',)); (x,) =
S._first_ngens(1)"
I recommend two things: (1) that you do not use names for the
generators of your number fields that look like indeterminates (x, y);
(2) that you do not reply on the pre-definition of 'x', but define it
yourself to be a generator of the polynomial ring over QQ which is
where the polynomials x^2+5 and x^2+7 really belong:
sage: x = polygen(QQ)
sage: S.<a> = NumberField(x^2+5)
sage: R.<b> = NumberField(x^2+7)
sage: a^2, b^2
(-5, -7)
John Cremona
>
> When defining R, x is not a polynomial variable, but an element of S. The
> line
> R.<y> = NumberField(x^2+7)
> really is
> R.<y> = NumberField(S(2))
>
> The only bug here might be that the error message is very confusing. It
> comes from
>
> sage: S(2).polynomial(QQ)
> ValueError: variable names must be alphanumeric, but one is 'Rational Field'
> which is not.
>
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