In the extension of an infinite field K by K(x,y), the intermediate field K(x) is Galois over K, but not stable (relative to K(x,y) and K). [See Exercise 9]

Data from exercise 9 

(a) If K is an infinite field, then K(x) is Galois over K. 

(b) If K is finite, then K(x) is not Galois over K.