This is an exercise problem from Hungerford Chapter 5.
Let $K$ be an infinite field. Prove that the intermediate field $K(x)$ of the extension $K\leq K(x,y)$ is Galois over $K$, but is not stable relative to $K(x,y)$ and $K$.
Here stable means: if $E$ is an intermediate field of the extension $K\subset F$, $E$ is said to be stable (relative to $K$ and $F$) if every $K$-automorphism $\sigma\in Aut_KF$ maps $E$ into itself.
This question was already asked here:$K(x)$ not stable relative to $K(x,y)$ and $K$, but the problem remains unanswered. Any help is appreciated.
We need $\sigma \in \mathrm{Aut}_K K(x,y)$. To specify a $\sigma$ it suffices to specify the images of $1$, $x$, and $y$. Try $$ \sigma : \begin{Bmatrix} 1 \mapsto 1 \\ x \mapsto y \\ y \mapsto x \end{Bmatrix} \text{.} $$ (Of course, we already know $$(K(x))(y) = K(x,y) \cong K(y,x) = (K(y))(x) \text{,} $$ so there should be o surprises showing this $\sigma$ is an automorphism fixing $K$ pointwise.) $K(x)$ is not mapped into $K(x)$. In fact, $\sigma(K(x)) \cap K(x) = K$ but $\sigma(x) = y \not \in K(x)$.