I'm trying to find the effect of $\sigma_a$ on square roots where $\sigma_a$ is defined as an automorphism on the Galois group of a cyclotomic field such that.
$$\sigma_a(\zeta_n)=\zeta_n^a$$
Where $\zeta_n$ is the nth root of unity. Note that $\sigma_a$ is multiplicative and additive. I've found that I can write the specific square roots I need as a sum of roots of unity i.e.
$$\sqrt{2N}=\sum_{b=0}^{2N-1} \exp(\frac{(2\pi i(2b^2 -N))}{8N})$$
where $N\in \mathbb{N}$ . Hence I would need to figure out how $\sigma_a(\sqrt{2N})$ looks. More specifically is there a way of saying what the square roots get mapped to as I've been told they should map back to another root of unity in the same set of roots of unity.
Also, would there be any way to write this out in Python?