Sizes of Quotient Rings of DVRs with Finite Residue Field If $R$ is a discrete valuation ring (DVR) with maximal ideal $\mathfrak{m}$ such that $R/\mathfrak{m}$ is finite, then all quotient rings of $R,$ namely $R/\mathfrak{m}^n$ for $n \in \mathbb{N},$ are finite. My question is:

Can we say anything about the sizes of $R/\mathfrak{m}^n$ for $n>1,$ if
  we know $\#(R/\mathfrak{m})$?

Many thanks!

We have a short exact sequence: $$0\to\mathfrak m/\mathfrak m^2\to R/\mathfrak m^2\to R/\mathfrak m \to 0.$$ Since $\dim_{R/\mathfrak m}\mathfrak m/\mathfrak m^2=1$ we get $\#(\mathfrak m/\mathfrak m^2)=\#(R/\mathfrak{m})$. It follows that $\#(R/\mathfrak m^2)=\#(R/\mathfrak{m})^2$.
Can you continue from here?
