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Jan 31, 2021, 3:45:28 PM1/31/21

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Dear sage-nt group!

I'm working on some algorithm using the reduction modulo p of polynomial maps defined over Z. I'm just wondering if it is possible to implement something like that for Gaussian Integers (just like in thread: https://math.stackexchange.com/questions/2160330/about-ring-of-gaussian-integers-modulo-n) Is something like that implemented in Sage? How can I define such a structure?

Best regards

Paweł Bogdan

Jan 31, 2021, 6:21:29 PM1/31/21

to sag...@googlegroups.com

Dear Paweł,

* Paweł Bogdan <pawel....@gmail.com> [2021-01-31 12:45:28 -0800]:

> [...]

if you are looking for a reduction map from the Gaussian Integers to a

residue field, the following might be more or less what you are trying

to do:

sage: Z.<I> = GaussianIntegers()

sage: R.<x> = Z[]

sage: p = 3

sage: red = R.hom(R.change_ring(Z.residue_field(prime=p)))

sage: g = R.random_element(); g

(-I - 7)*x^2 + x + 4*I - 7

sage: red(g)

(2*Ibar + 2)*x^2 + x + Ibar + 2

sage: p = Z.ideal(2).factor()[0][0]

sage: red = R.hom(R.change_ring(Z.residue_field(prime=p)))

sage: red(g)

x + 1

Is that what you had in mind?

julian

* Paweł Bogdan <pawel....@gmail.com> [2021-01-31 12:45:28 -0800]:

> [...]

> Is something like that implemented in Sage? How can I define such a

> structure?

I am not sure I understood what you are trying to achieve exactly. But
> structure?

if you are looking for a reduction map from the Gaussian Integers to a

residue field, the following might be more or less what you are trying

to do:

sage: Z.<I> = GaussianIntegers()

sage: R.<x> = Z[]

sage: p = 3

sage: red = R.hom(R.change_ring(Z.residue_field(prime=p)))

sage: g = R.random_element(); g

(-I - 7)*x^2 + x + 4*I - 7

sage: red(g)

(2*Ibar + 2)*x^2 + x + Ibar + 2

sage: p = Z.ideal(2).factor()[0][0]

sage: red = R.hom(R.change_ring(Z.residue_field(prime=p)))

sage: red(g)

x + 1

Is that what you had in mind?

julian

Feb 6, 2021, 4:10:12 PM2/6/21

to sage-nt

Dear Julien!

Yes, this is exactly what I meant. Thank you very much!

Best regards

Paweł

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