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

Question

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!

Answer 1

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.