I have a question about the computation of characters of $\mathbb{F}_{2^n}$ in arXiv:quant-ph/0410155. My question might be trivial but I'm not very familiar with details in the theory of finite fields (since I'm a physicist working in quantum computing). I want to understand the construction of the mutually unbiased basis sets for qubits, which they propose in the paper - in particular their example for three qubits in section 3.3. My problem is the computation of the characters.
The field of which they compute the characters is described as: "The field $\mathbb{F}_d$ can be represented as the field of equivalence classes of polynomials whose coefficients belong to $\mathbb{Z_p}$." where $d=p^n$ and in my case I need $p=2$.
To illustrate my problem, let's have a closer look at the example for three qubits. The field is $\mathbb{2^3} = \mathbb{F}_8$ and it has the elements
$$ \{ 0,1,\alpha, \alpha^2, \alpha+1, \alpha^2+\alpha, \alpha^2+\alpha+1, \alpha^2+1 \}$$
with the relations
$$ \alpha^3=\alpha+1, \alpha^4=\alpha^2+\alpha, \alpha^5=\alpha^2+\alpha+1, \alpha^6=\alpha^2+1, \alpha^7=1 $$
The characters for each element will be $\pm 1$. They say they are (and I want to understand in detail how to retrieve them)
$$ \chi(0)=1, \chi(\alpha)=1, \chi(\alpha^2)=1, \chi(\alpha^3)=-1, \chi(\alpha^4)=1, \chi(\alpha^5)=-1, \chi(\alpha^6) = -1, \chi(\alpha^7)=-1 $$
and they can be computed via $$ \chi(\theta)=\exp\Bigl( \frac{2\pi i}{p} \mathrm{tr}(\theta) \Bigr). $$
The trace of those elements is defined as
$$ \mathrm{tr}(\theta) = \theta + \theta^p + +....+ \theta^{p^{n-1}}. $$
It is absolutely clear to me how to compute the characters $\chi(0)=1$ and $\chi(\alpha^7)=\chi(1)=-1$, since you can simply plug in numbers into the above formula. The problem for me arises with the other characters. Since the characters are additive (hence $\chi(\theta_1+\theta_2)=\chi(\theta_1)\chi(\theta_2)$ holds) one can rewrite the remaining characters as
$$\chi(\alpha^3) = \chi(\alpha)\chi(1)$$ $$ \chi(\alpha^4) = \chi(\alpha^2) \chi(\alpha) $$ $$ \chi(\alpha^5) = \chi(\alpha^2)\chi(\alpha)\chi(1) $$ $$ \chi(\alpha^6) = \chi(\alpha^2)\chi(1) $$
Thus, the problem reduces to finding the characters $\chi(\alpha)$ and $\chi(\alpha^2)$. But I do not know how to retrieve a specific number ($\pm 1$) from the formula as soon as I have the character in terms of $\alpha$'s. I was already thinking that the set of characters is not uniquely determined (such that one can consistently choose them according to the above formulas), but I'm pretty sure that this is just an idea which stems from my lack of knowledge about this stuff. How can I explicitly compute the characters if I want a character of an element which is in terms of the primitive element $\alpha$?