# Thread: proving that an extention is cyclic

1. ## proving that an extention is cyclic

1. An extension L/K is cyclic if and only if it is Galois and its Galois group is cyclic. Assume
that L/K is cyclic and F is a extension of K such that K ⊆ F ⊆ L. Prove that L/K is cyclic and F/K is cyclic by showing that the extention is finite, normal, and its galois group is cyclic

2. Originally Posted by maroon_tiger
1. An extension L/K is cyclic if and only if it is Galois and its Galois group is cyclic. Assume
that L/K is cyclic and F is a extension of K such that K ⊆ F ⊆ L. Prove that L/K is cyclic and F/K is cyclic by showing that the extention is finite, normal, and its galois group is cyclic
The result that you need to use here is the fundamental theorem of Galois theory . The group $\displaystyle \mbox{Gal}(L/K)$ is a cyclic group since $\displaystyle L/K$ is a cyclic extension. This means any subgroup is cyclic, in particular, $\displaystyle \mbox{Gal}(L/F)$, thus $\displaystyle L/F$ is cyclic extension. Since $\displaystyle \mbox{Gal}(L/F)\triangleleft \mbox{Gal}(L/K)$ it means $\displaystyle F/K$ is a normal extension by Galois theory. Finally, $\displaystyle \mbox{Gal}(L/K)/\mbox{Gal}(L/F) \simeq \mbox{Gal}(F/K)$, since a factor group of a cyclic group is cyclic this means $\displaystyle F/K$ is also cyclic.