1 | initial version |

Checking the documentation for `NumberField`

(you can use "`NumberField?`

" to print the documentation from within sage), I found an optional argument `embedding`

which specifies the embedding into another field -- so you have to specify *which* `zeta6`

you mean:

Here are the roots over `CC`

:

```
sage: cyclotomic_polynomial(6).base_extend(CC).roots()
[(0.500000000000000 - 0.866025403784439*I, 1), (0.500000000000000 + 0.866025403784439*I, 1)]
```

Here we use the first root for our embedding:

```
sage: K.<a>=NumberField(cyclotomic_polynomial(6), embedding=cyclotomic_polynomial(6).base_extend(CC).roots()[0][0])
sage: R=K.maximal_order()
sage: A=DirichletGroup(7)
sage: character=A[1]
sage: character(3) in R.fractional_ideal(a)
True
```

Coercing to `CC`

shows what embedding we're using:

```
sage: CC(a)
0.500000000000000 - 0.866025403784439*I
```

You can also use the other root of the cyclotomic polynomial for your embedding:

```
sage: K.<a>=NumberField(cyclotomic_polynomial(6), embedding=cyclotomic_polynomial(6).base_extend(CC).roots()[1][0])
sage: CC(a)
0.500000000000000 + 0.866025403784439*I
```

You may also be interested in `CyclotomicField`

, which I came across while I was looking at the documentation for `NumberField`

-- it seems to construct the `nth`

cyclotomic field with canonical embedding to `CC`

automatically...

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