# Cyclotomic fields - displaying powers of zeta

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^5
-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)
zeta^2
```

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>`

.

`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.Basically I want a structure where I have both

`zeta^2`

(or at least for display) and ability to factor in the polyring.Then work modulo

`zeta^3 - 1`

while you don't do factoring; and when you do, embed the argument into the cyclotomic field first.