1 | initial version |

It is fairly straightforward once you know Sage standards

```
sage: K = CyclotomicField(9)
sage: O = K.ring_of_integers()
sage: zeta9 = O.gen(1)
```

At this point you have three objects defined in the console/notebook: the cyclotomic field **K**, its ring of integers **O** and the generator **zeta9**. Now you can defined ideals and quotients as follows.

```
sage: I = O.ideal(3*zeta9^2 + 2*zeta9^3 + 5)
sage: R = O.quotient(I, 'a')
```

note: I am not sure why, but the arguemnt 'a' is mandatory in the method **quotient**.

To check equality modulo **I** just do it in the quotient

```
sage: R(3*zeta9^2 + 7) == R(2 - 2*zeta9^3)
True
sage: R(zeta9) == R(2)
False
```

Vincent

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