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 $K[X,Y]=\mathbb{F}_p(X,Y) \subseteq \mathbb{F}_{p^2}(X,Y) =L\,.$