Let $(R,m)$ be a regular local ring of dimension $d$ and char $p>0.$ Let $F^e:R\longrightarrow R$ defined by $r\longrightarrow r^{p^e}$be the Frobenius map.


How to compute $l(R/m^{[p^e]})?.$


I know the answer is $p^{ed}$ but I do not know how to prove it.



Length of a module and Frobenius map
This is false. The noetherian local ring $R = 
\mathbb{F}\_3[[X,Y]]/(Y^2 - X^3)$ has dimension one, and if $x,y$ are the images of $X,Y$ in $R$ then the sequence
$$
R \supseteq (x,y) \supseteq (x^2,y) \supseteq (y) \supseteq (y^2) = \mathfrak{m}^{[3]}
$$
shows that $R / \mathfrak{m}^{[3]}$ does not have $R$-length $3$.


