The integral closure of a power series ring over a field

The integral closure of a power series ring over a field

Let $k$ be a field of characteristic $p$ and $K$ the field of fractions of
the formal power series ring $k[[X_1,\cdots,X_n]]$. Let $L$ be a finite
purely inseparable field extension of $K$ , then there exists an integer
$m$ such that $L\subseteq K^{{1/q}}$ with $q=p^m$. Set $k^{'}=k^{1/q}$ and
$Y_i=X_i^{1/q}$, let $K^{'}$ and $k^{'}((Y_1,\cdots,Y_n))$ denote the
field of fractions of $k[[Y_1,\cdots,Y_n]]$ and $k^{'}[[Y_1,\cdots,Y_n]]$
respectively, then $K^{'}\subseteq L(Y_1,\cdots,Y_n) \subseteq
k^{'}((Y_1,\cdots,Y_n))$. The integral closure $A$ of
$k[[Y_1,\cdots,Y_n]]$ in $L(Y_1,\cdots,Y_n)$ is just
$k^{'}[[Y_1,\cdots,Y_n]] \cap L(Y_1,\cdots,Y_n)$. Now the question is:
Why the maximal ideal $\mathbb{m}$ of $A$ is generated by $Y_1,\cdots, Y_n$?
From $k[[Y_1,\cdots,Y_n]] \subseteq A \subseteq k^{'}[[Y_1,\cdots,Y_n]]$
and A is integral over $k[[Y_1,\cdots,Y_n]]$, I can only konw
$(Y_1,\cdots,Y_n)$ is $\mathbb{m}$-primary.