# Morphism of Curves: Is the exponent of inseparability bounded by the degree?

Let $X,Y$ be complete, nonsingular Curves over an algebraically closed field $k$. Let $f:X\to Y$ be a finite Morphism, such that $K(X)/K(Y)$ is a purley inseparable extension. Let $p^r$ bet the degree of $f$. Then Hartshorne in IV.2.5 just states that $K(X)\subset K(Y)^{1/p^r}$, but I could only find, that for all $a\in K(X)$ there exitsts an $m$, such that $a^{p^m}\in K(Y)$.

• How have you used that the degree is $p^r$? Your conclusion holds for any purely inseparable map independent of the degree. Apr 13, 2017 at 15:59
• I haven't used that, and I know that that is a general fact, but it seems my question should somehow easly follow from the fact. Apr 13, 2017 at 16:10
• May be you should try $r=1$ case and see why this is true. Apr 13, 2017 at 17:06
• Well for $r=1$ we have that the minimalpolynomial for any $a\in K(X)$ is of the form $(x-a)^{p^m}$ but as $p^m <= p$ iff $m=1$, but this does not work for any $r>1$ as then the minimal polynomial could be of any degree $p^n$ where $n<r+1$, to do the same we would need to have that $n|r$ but I can`t see why that has to be the case. Apr 13, 2017 at 18:09