My question arises in the context of the Shtuka correspondence, but there is only pure abstract algebra involved.
In the book "Basic structures of Function Field Arithmetic" by David Goss, in chapter $6$ which deals with the Shtuka correspondence, there is Proposition $6.2.2$.
which makes the following statement:
Let $L$ be a field of any characteristic and let $\sigma$ be an automorphism of $L$ of infinite order. Let $L_0$ be the fixed field of $L$
with respect to $\sigma$.
Let $X$ be a complete integral curve over $L_0$ and let $\overline{X}$ be its base change to $L$. Assume that $\overline{X}$ is also irreducible. Then $\overline{X}$ is actually integral aswell.
The proof starts with the assertion that one only needs to show that $L/{L_0}$ is separable. This is clear to me, because one only needs to prove that $\overline{X}$ is still reduced, and this is just Lemma 10.43.1.
But then the proof goes as follows:
Suppose that there is a sequence $\alpha_1,\cdots,\alpha_n \in L_0^{\frac{1}{p}}$ whiche are linearly independent over $L_0$ but dependent over $L$ with minimal $n$. Then ... and he concludes a straightforward contradiction.
I have no idea why linearly dependence/independence should have anything to do with the fact that $L/{L_0}$ is separable in the first place. Why should this be a proof of the statement? I understand the way the contradiction is obtained, but I don't understand, why it proves separability
For a field $k$ of characteristic $p>0$ the notation $k^{\frac{1}{p}}$ denotes the extension field of $k$ containing all roots $\sqrt[p]{x},x\in k$.
I am happy if somebody could elaborate on this. I don't see why this should be a proof of the proposition.
Thank You for Your help, appreciate, SDIGR