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?

 

[tacman]

There is not necessarily a trick to determine whether Gal(F/G) is isomorphic to the Klein 4 or a cyclic group of order 4. However, there are some properties that can help in making this determination.

Firstly, if all the roots of g(x) are distinct and the degree of g(x) is 4, then Gal(F/G) is isomorphic to the cyclic group of order 4. This is because in this case, all the roots of g(x) will be contained in the splitting field F, and the Galois group will be generated by the automorphism that maps each root to its consecutive root.

On the other hand, if the degree of g(x) is greater than 4, then it is possible for Gal(F/G) to be isomorphic to the Klein 4 group. This is because in this case, g(x) may have repeated roots that are not contained in the splitting field F. In this case, the Galois group will be generated by two automorphisms, one that maps each root to its consecutive root, and another that maps each root to its negative.

Therefore, in order to determine whether Gal(F/G) is isomorphic to the Klein 4 or a cyclic group of order 4, one would need to consider the roots of g(x) and their multiplicities. If all the roots are distinct and the degree is 4, then it is cyclic. If there are repeated roots or the degree is greater than 4, then it is the Klein 4 group.