Cyclotomic fields - displaying powers of zeta

asked 2024-02-18 20:39:43 +0200

kejv2 gravatar image

Is there any way how to avoid expressing $\zeta^{n-1}$ in terms of lower powers of $\zeta$ when working with cyclotomic fields?

K.<zeta> = CyclotomicField(3)
-zeta - 1

I would like to show $\zeta^2$ in this case.

I know that I can do this instead:

K.<zeta> = QQ[]
zeta^5 % (zeta^3 - 1)

But the problem is that eventually, I want to work with a polynomial ring over K and perform a factorization there, and that is not possible in QQ[].<x,y>.

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zeta^5 % (zeta^3 - 1) is not the same as computing zeta^5 in CyclotomicField(3) as the latter is done modulo cyclotomic polynomial zeta^2 + zeta + 1. So it's unclear what you want.

Max Alekseyev gravatar imageMax Alekseyev ( 2024-02-18 22:55:12 +0200 )edit

Basically I want a structure where I have both zeta^2 (or at least for display) and ability to factor in the polyring.

kejv2 gravatar imagekejv2 ( 2024-02-19 08:28:26 +0200 )edit

Then work modulo zeta^3 - 1 while you don't do factoring; and when you do, embed the argument into the cyclotomic field first.

Max Alekseyev gravatar imageMax Alekseyev ( 2024-02-19 12:14:14 +0200 )edit