Galois and Klein 4: Isomorphic or Cyclic?

[morganjp]

suppose a +ve degree polynomial g(x) in G[x] with F splits over G and no irr. factor has repeated root. then if [F:G]=4, we know the size of Gal(F/G) is also 4. so it's either isomorphic to the Klein 4 or cyclic group of order 4. is there any trick to tell it's one and not the other?

 

[morphism]

The Galois group will permute the roots. Can you find a permutation of order 4?