# Evaluating polynomials in Z[x] at roots of unity.

Suppose that I have a polynomial $p(x)$ with integer coeffcients, then I can easily evaluate this at some integer to get an exact answer. That is, I can write something like

```
sage: p=1+x+2*x^2+x^3+x^4
sage: p(2)
35
```

But If I instead want to evaluate this polynomial at a root of unity (I take $w$ to be a primitive $5$:th root of unity in the example below but any root of unity would do), then I may write say

```
sage: w = e^(2*pi*i/5)
sage: p(w)
1/256*(sqrt(5) + I*sqrt(2*sqrt(5) + 10) - 1)^4 + 1/64*(sqrt(5) + I*sqrt(2*sqrt(5) + 10) - 1)^3 + 1/8*(sqrt(5) + I*sqrt(2*sqrt(5) + 10) - 1)^2 + 1/4*sqrt(5) + 1/4*I*sqrt(2*sqrt(5) + 10) + 3/4
```

Which is a correct answer but written in a very complicated form. Alternatively, I can write

```
sage: K.<z> = CyclotomicField(5)
sage: w=CC(z)
sage: p(w)
-0.809016994374947 + 0.587785252292473*I
```

which is an approximation of the correct answers. In the above case, we actually have $p(w)=w^2$ and I am wondering if there is an easy way to always obtain such an answer. For a general $p(x)$ with integer coefficients and $w$ that is a primitive $n$:th root of unity, we can write $p(w)$ on the form

$$a_0+a_1 w + a_2 w^2 + \dots +a_{n-1} w^{n-1}$$

where the $a_i$:s are integers. Is there a good way to get the $p(w)$ on this form in Sage?

By the way, the degree of $w$ is $\varphi(n)$ in general, so there is an expansion with fewer powers.