The degree of a particular purely inseparable extension

1. Mar 28, 2010

eof

This one step in a long problem (an example in algebraic geometry cast in another language) that I've condensed below (p is the characteristic):

Let $$\overline{k}$$ be an algebraic closure of $$k$$ and let $$L=\overline{k}(Y)$$. Consider the monic polynomial

$$H(x)=x^s-(Y^p+Y^{-t})$$

in $$L[x]$$. Write $$s=p^ms'$$ for some integer $$s'$$ prime to $$p$$ and some $$m\geq 0$$. Let $$x_0$$ be a root of $$H(x)$$ in an algebraic closure $$\overline{L}$$ of $$L$$. Show that the field $$L(x_0)$$ is an inseparable degree $$p^m$$ extension of $$L(x_1)$$ where $$x_1=x_0^{p^m}$$.

Ok, so proving that the extension is inseparable is trivial. It then follows that the minimal polynomial of $$x_0$$ over $$x_1$$ has to be of the form $$(X-x_0)^{p^n}=X^{p^n}-x_0^{p^n}$$. This shows that $$x_0^{p^n}\in L(x_1)$$. Unfortunately, this only gives that $$n\leq m$$, but it does not show equality. Does anyone have any ideas on how to approach this (I spent about 2 hours yesterday thinking about this...).