# Show that $(L : K), (Z : K)$ and $(L : Z)$ are all Galois extensions, but that $Z$ is not a stable intermediate field of $(L : K)$.

Let $K$ be an infinite field and let $Z = K(x)$ and $L = K(x,y)$ with variables $x,y$.

Show that $(L : K), (Z : K)$ and $(L : Z)$ are all Galois extensions, but that $Z$ is not a stable intermediate field of $(L : K)$.

I have shown that if $K$ is an infinite field then $(Z : K)$ is a Galois extension.

A field extension $(L : K)$ is called a Galois extension, if $F(Aut(L/K)) = K$, i.e., if for any $a \in L- K$ there is an automorphism of $L$ which leaves $K$ pointwise fixed, but actually moves $a$.

$Aut(K(X)/K) = \{\phi \mid \phi : K(X) \to K(X) , \phi(k) = k \ \ \forall k \in K\}$ where $\phi$ is an automorphism.

Aut are of the form $\frac{ax+b}{cx+d}$.....can we conclude from here that for two variables we have $\frac{ax+by+cxy+d}{ex+fy+gxy+h}$

Can someone help in the problem with some hints?

Thank You!

• What are the elements of $Gal(K(x)/K)$ ? Commented Jan 9, 2017 at 3:39
• Since $Z/K$ and $L/K$ are not algebraic you need to make it clear what you mean by "Galois" Commented Jan 9, 2017 at 3:43
• You didn't answer : what is $Aut(K(x)/K)$ ? Then what part of $Aut(K(x,y)/K)$ can you construct ? Commented Jan 9, 2017 at 4:13
• Aut(K(x)/K) is the set of all automorphisms from K(x) to K(x) which which keep K fixed Commented Jan 9, 2017 at 4:16
• We know the definition, I'm asking what did you find for this group (and also what do you get for its fixed field) Commented Jan 9, 2017 at 4:19

For this definition of Galois extensions which does not require algebraicness of the extension, you say you've shown that $$(K(x):K)$$ is Galois if $$K$$ is infinite. Applying this again to $$K(x)$$, we have $$K(x,y)=K(x)(y)$$ is Galois over $$K(x)$$. Now we wish to prove $$K(x,y)$$ is Galois over $$K$$. By a well-known result about nested Galois extensions, it suffices to prove every $$K$$-automorphism of $$K(x)$$ extends to one of $$K(x,y)$$. But if $$x\mapsto \frac{ax+b}{cx+d}$$ is one such automorphism, then $$(x,y)\mapsto (\frac{ax+b}{cx+d},y)$$ is one such extension.
The only thing left to prove now is that the middle field isn't stable. But this is easy, take $$(x,y)\mapsto (y,x)$$. This is a $$K$$-auto of $$K(x,y)$$ that doesn't send $$K(x)$$ to itself.