Normal field extension


how can I show that a field extension is normal?

Here is a concrete example:
[tex]L|K [/tex] is normal, whereas [tex]L=\mathbb F_{p^2}(X,Y) [/tex] and [tex] K= \mathbb F_p(X^p,Y^p) [/tex].
[tex] p [/tex] is a prime number of course.

I have to show that every irreducible polynomial in [tex]K[X,Y][/tex] that has a root in [tex]L[/tex] completely factors into linear factors over [tex]L[/tex].

But this is not simply in my case, because elements in [tex]K[X,Y]=\mathbb F_p(X^p,Y^p)[X,Y] [/tex] has the form:
[tex] \frac{g(x,y)}{h(x,y)}, \quad h(x,y)\neq 0, \quad g,h \in K[X,Y] [/tex]


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