# multiple algebraic extension

I'm looking for a way to formulate a number field which can express corners of a given regular $n$-gon exactly, but which might be faster than `AA`

. I need to be able to check signs of elements of that number field. Here is what I tried:

```
n = 17
z = QQbar.zeta(n)
ca = z.real()
sa = z.imag()
QQ[ca, sa]
```

However I get the error message

```
ValueError: defining polynomial (x^16 - 17/4*x^14 + 119/16*x^12 -
221/32*x^10 + 935/256*x^8 - 561/512*x^6 + 357/2048*x^4 - 51/4096*x^2 +
17/65536) must be irreducible
```

How can I avoid this?