# Computations with complex algebraic numbers?

I am trying to perform some Newton-type computations: `z-p(z)/p'(z)`

where `p(z)`

is a polynomial over the field of algebraic numbers.

For example, this shows the sort of thing I'm trying to do:

```
p = (z^2+1)*(z^2+4)
q = p/(z-I)
pd = diff(p,z)
qd = diff(q,z)
sol = solve(qd=0,solution_dict='true')
a0 = sol[0][z]
b0 = a0-p.subs(z=a0)/pd.subs(z=a0)
```

However, this doesn't work because the expression `z-I`

is not recognized as a factor of `p`

. So I might try defining
`p`

as a polynomial over the field `QQbar`

:

```
R.<z> = PolynomialRing(QQbar)
p = (z^2+1)*(z^2+4)
q = p.quo_rem(z-I)[0]
pd = p.differentiate(z)
qd = q.differentiate(z)
```

The trouble now is that I can only seem to solve `qd`

over `CC`

: that is, in numerical form. Both the commands

```
qd.roots()
qd.factor()
```

produce numerical roots; I can't use `solve`

on `qd`

to obtain closed-form expressions for the roots. I've also tried the first method above prefaced with `z=QQbar['z'].0`

but to no avail.

Is there some way of doing this entire computation over `QQbar`

, that is, with exact complex numbers instead of numerical approximations?