primitive k-th root of unity

[mac8090]

For a given k, is it possible to instantly get an k-th root of unity
in sage without making extra fields, or by using e^(2*pi*I/k)?

On a similar note, anybody know why I can't get sage to equate e^
(theta*I) == cos(theta) + I*sin(theta) ?

[davidloeffler]

On Jul 14, 11:37 am, mac8090 <bonzerpot...@hotmail.com> wrote:
> For a given k, is it possible to instantly get an k-th root of unity
> in sage without making extra fields, or by using e^(2*pi*I/k)?

I'm curious why you are so opposed to creating a number field.
Basically, there are three one-liners you can use:

CyclotomicField(k).gen() -- creates a number field element.
CC.zeta(k) -- creates an inexact, fixed-precision approximation.
QQbar.zeta(k) -- a clever hybrid, which behaves like an element of CC
but also remembers that it's algebraic and can calculate itself to
arbitrary precision if you ask it to.

David

[David Joyner]

On Tue, Jul 14, 2009 at 6:37 AM, mac8090<bonzer...@hotmail.com> wrote:
>
>
> For a given k, is it possible to instantly get an k-th root of unity
> in sage without making extra fields, or by using e^(2*pi*I/k)?

I'm a bit confused by your question. If you mean k-th roots of unity in
the complex field CC then

sage: z = e^(2*pi*I/7)
sage: z^7
1

works for me. If youmean the kth roots of unity in some other field
(after all, every field contains 0 and 1) then you should specify the
field to determine where you want the roots of x^k-1=0 to lie.

>
> On a similar note, anybody know why I can't get sage to equate e^
> (theta*I) == cos(theta) + I*sin(theta) ?

I don't know. Sage uses Maxima. Does maxima know Euler's formula?


> >
>

[Laurent]

>> On a similar note, anybody know why I can't get sage to equate e^
>> (theta*I) == cos(theta) + I*sin(theta) ?
>>
>
>
> I don't know. Sage uses Maxima. Does maxima know Euler's formula?
>
>

I suppose that Sage knows Euler's formula because

sage: var('x')
x
sage: real_part(e^(I*x))
e^(-imag_part(x))*cos(real_part(x))

but :

sage: var('x')
x
sage: assume(imag_part(x)==0)
sage: real_part(e^(I*x))
e^(-imag_part(x))*cos(real_part(x))

seems strange to me.
Does someone know how to make Sage understand that x is real and that
e^(i*x) = cos(x)+I*sin(x) ??

Good afternoon
Laurent