Proving Normal Field Extensions with an Example | Field Extension Normality

[brian_m.]

Hi,

how can I show that a field extension is normal?

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

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

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

Bye,
Brian

 

[fresh_42]

I think we have $X^p\equiv X\, , \,Y^p\equiv Y$ which makes $K[X,Y]=\mathbb{F}_p(X,Y) \subseteq \mathbb{F}_{p^2}(X,Y) =L\,.$