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I am trying to find the exact radicals of the sine and cosine of (m/n)*pi. I have been using sympy to do this and made this pretty good script: ...

For the following, fix $m\geq 2$, and let $a_{0},\dots,a_{m-1}\in\mathbb{Z}$ be such that $\sum_{k=0}^{m-1}a_{k}=0$. I would like to find the exact conditions on the $a_{k}$ so that the following ...

Let $\zeta$ be a primitive $N$-th root of unity and $\Gamma \subset (\mathbb{Z}/N)^\times$ a subgroup. Let $|\Gamma|$ be the cardinality of $\Gamma$ and consider the linear map $M_\Gamma\colon \mathbb{...

Let $n \in \mathbb{N}$ and $\zeta$ be a primitive $n$-th root of unity. I want to know for which $n$ the element $1+2(\zeta+\zeta^{-1})$ is a unit in the ring of integers of $\mathbb{Q}[\zeta]$. Can ...