All Questions

Let $n$ be a positive integer and $\zeta$ be a primitive $n$th root of unity. It is not hard to show that \begin{align*} \sum_{k=1}^{n-1}\frac{\zeta^k}{1-\zeta^k}=\frac{1-n}{2}. \end{align*} Since $\...

Pointed out on famous disbelief, I know now that there is an embedding $\iota_n \colon {\Bbb F}_p[[T_1,...,T_n]] \hookrightarrow {\Bbb F}_p[[X,Y]]$ for any $n \leq \infty$. Then I would like to ask ...

Let $K[[X,Y]]$ be a two-variables formal power series ring over a field $K$. Consider a sub-ring $\iota \colon A \subset K[[X,Y]]$. Q. Is A coherent? $\quad$ Or is it automatic that $\iota$ is ...

$\newcommand{\Qbar}{\overline{\mathbb{Q}}}$ We define a modular function to be a meromorphic modular form of weight 0 for some subgroup (not necessarily congruence) $\Gamma\le\text{SL}_2(\mathbb{Z})$ ...

Let ${\Bbb F}_2[[T]]$ be a c.d.v.r over ${\Bbb F}_2$. We consider the automorphism $\sigma$ of ${\Bbb F}_2[[T]]$ such that $\sigma \colon T \mapsto T + c_2T^2 + c_3T^3 + \cdots$ with $c_i \in {\Bbb F}...