My question relates to p. 147 of Serre's Linear Representations of Finite Groups, where he is setting up the definitions relevant to Brauer character theory.
Having fixed an algebraically closed field $K$ of characteristic $p$, we have a finite group $G$ and the set $G^p$ of $p$-regular elements of $G$, i.e. those elements whose order $\left|g\right|$ is coprime to $p$.
Assume $K$ is complete with respect to a discrete valuation $\nu:K^\times\to \mathbb{Z}$ with valuation ring $A=\{x\in K\mid \nu(x)\geq 0\}$. By general results about discrete valuation rings, $A$ has a unique maximal ideal $\mathfrak{m}=\{x\in K\mid \nu(x)\geq 1\}$, so the quotient $A/\mathfrak{m}$ is a field $k$, called the residue field of $K$.
Let m'=\mathop{\mathrm{lcm}}_{g\in G^p}\left|g\right|.
Q0: Is $\mathrm{char}(k)=p$?
Q1: Why must $K$ contain the group $\mu_K$ of $m'$th roots of unity? (I understand this is to do with $K$ being algebraically closed, but can't the fact that $K$ has characteristic $p$ screw things up? What if m'>p?)
Q2: What is the map $K\to k$ given by reduction modulo $\mathfrak{m}$ and why must this be an isomorphism $\mu_K\to \mu_k$, where $\mu_k$ is the group of $m'$th roots of unity in $k$? And again, why must $\mu_k\subseteq k$?
Q3: For $\lambda\in \mu_k$, what does he mean by $\overline{\lambda}$, the "element of $\mu_K$ whose reduction modulo $\mathfrak{m}$ is $\lambda$"?
Q4: Is there a nice example of all of this theory using a particularly simple field of characteristic $p$ with a nice valuation that I would be familiar with and could do some basic computations with?
That's quite a lot of questions, I should leave it there. I hope someone can help me out -- but it definitely just helps to write it out anyway!
Cheers,
Clinton