Let $\mathbb{F}_p$ be the finite field of characteristic $p>0$. Consider the ring of power series $R=\mathbb{F}_p[[x_1,x_2, \cdots, x_n]]$ and its free subring $S=\mathbb{F}_p[[x_1^{p^{i_1}}, x_2^{p^{i_2}}, \cdots, x_n^{p^{i_n}}]]$ for some positive integers $i_1, i_2, \cdots, i_n$.
What would be the rank of $R$ over $S$ ?
If we consider just one variable then $\mathbb{F}_p[[x]]$ over $\mathbb{F}_p[[x^{p^i}]]$ has a basis $1,x,x^2, \cdots, x^{p^i-1}$ and so the rank is $p^{i}$.
Then $R$ over $S$ has rank $p^{i_1} \cdot p^{i_2} \cdots p^{i_n}=p^{i_1+i_2+\cdots+i_n}$.
Is that all ?
Any help please
