# Normal field extension

1. Dec 13, 2009

### 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