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Math Help - proving that an extention is cyclic

  1. #1
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    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
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    Quote Originally Posted by maroon_tiger View Post
    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 \mbox{Gal}(L/K) is a cyclic group since L/K is a cyclic extension. This means any subgroup is cyclic, in particular, \mbox{Gal}(L/F), thus L/F is cyclic extension. Since \mbox{Gal}(L/F)\triangleleft \mbox{Gal}(L/K) it means F/K is a normal extension by Galois theory. Finally, \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 F/K is also cyclic.
    Last edited by ThePerfectHacker; April 25th 2008 at 09:22 AM.
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