Dual of an ideal

[Cindy]

Hi,

How can I calculate the dual of an ideal using sage?

Thanks.

Cindy

[David Loeffler]

What exactly do you mean by the dual of an ideal? Do you mean dual
with respect to the trace pairing, so the dual of the ideal (1) is the
inverse different?

David

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[vijay sharma]

Cindy,

Could you elaborate little more, what is precisely you need.

Regards,

Vijay

[Cindy]

Hi, David,

Yes, that's what I mean. Can I find it using sage?

Thanks.

Cindy

[Cindy]

Hi, Vijay,

Let K be a number field and O_k be its ring of integers. Given an ideal J of O_k, I want to find the dual of J, which is defined as the O_k-module:

J^*={x\in K| Tr(xJ)\subset Z}.

Thanks.

Cindy

[Cindy]

Hi,

BTW, the ideals I am dealing with are ideals of the ring of integers of a number field.

Cindy

On Tuesday, September 4, 2012 3:12:25 PM UTC+8, David Loeffler wrote:

[David Loeffler]

On 5 September 2012 02:41, Cindy <cindy42...@gmail.com> wrote:
> Hi, David,
>
> Yes, that's what I mean. Can I find it using sage?
>
> Thanks.
>
> Cindy

sage: K.<z> = NumberField(x^3 - x + 17)
sage: I = K.primes_above(17)[0]
sage: K.trace_dual_basis(I.basis())
[4/132583*z^2 + 6/7799*z + 2597/132583, -153/7799*z^2 - 2/7799*z +
102/7799, -6/7799*z^2 - 153/7799*z + 4/7799]

hth, David

[Cindy]

Hi, David,

Could you please explain a little bit about the code?

For the example you use, it seems I is an ideal above 17, what does [0] mean?

In the end do we get a basis of the dual of I? Why do we need to put I.basis() in the bracket of trace_dual_basis?

Thanks a lot.

Cindy

[David Loeffler]

On 5 September 2012 09:34, Cindy <cindy42...@gmail.com> wrote:
> Hi, David,
>

> Could you please explain a little bit about the code?

Sure, but you should make a little effort to play with it yourself for
a bit first.

> For the example you use, it seems I is an ideal above 17, what does [0]
> mean?

The command K.primes_above(...) returns a list of the prime ideals
above the given rational prime. The [0] selects the first (zeroth?)
from the list. So yes, I is an ideal above 17 which I am just using as
an example (any number field ideal, except the zero ideal, would work
here). There are lots of examples like this in the Sage documentation.

> In the end do we get a basis of the dual of I?

Yes, that's the whole point of the exercise :-). Did you read the
documentation for "trace_dual_basis"? You should know that you can get
documentation on any method of any Sage object by typing its name then
?, e.g.

sage: K.trace_dual_basis?

will tell you lots more about this method.

> Why do we need to put
> I.basis() in the bracket of trace_dual_basis?

Because trace_dual_basis takes a list of generators as its argument --
it can calculate the trace dual of any Z-submodule of K, it needn't be
an ideal.

Regards, David

[Cindy]

Hi, David,

Thanks a lot. I tried trace_dual_basis? to find out the meaning. I didn't realize I should use K.trace_dual_basis?

Thanks. :)

Cindy