Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Please help! Field extension problem

  1. Feb 11, 2009 #1
    if K/E is a quadratic extension and field F is contained in K
    such that FE=K and [K:F] is finite,
    how do I give a non-example to show
    [F: F intersects E] might not be 2?

    Thanks a lot!
     
  2. jcsd
  3. Feb 12, 2009 #2
    Very EASY!
    Do you have Galios theory in your hands?

    Well consider [tex]K[/tex] the splitting field of [tex]x^3 - 2[/tex] over [tex]\mathbb Q[/tex].

    By Galois theory you should know the lattice of inter-fields between [tex]Q[/tex] and
    [tex]K[/tex] is isomorphic to the lattice subgroup of the group [tex]\mathbb S_3[/tex]
    of the permutations on 3 elements.

    Now this lattice as a unique subgroup on 3 elements and 3 distinct subgroups of 2 elements. Choose two distinct of these and call them [tex]G_1, G_2[/tex].
    Let's call [tex]e[/tex] the trivial subgroup (just one element: the identity permutation).

    Call [tex]\prime[/tex] the Galois corrispondence and you have the fields

    [tex]K = e \prime[/tex]
    [tex]E = G_1 \prime[/tex]
    [tex]F = G_2\prime[/tex]
    and [tex]E \cap F = (G_1\cdot G_2)\prime = \mathbb S_3\prime = \mathbb Q[/tex].

    You have [tex][K:E] = [G_1:e] = 2[/tex] and [tex]K/E[/tex] is a quadratic extension
    You have [tex][K:F] = [G_2:e] = 2[/tex] and [tex]K/F[/tex] is a finite extension
    You have [tex]F\cdot E = (G_2 \cap G_3)\prime = e\prime = K[/tex]
    You have [tex][F:E\capF] = [\mathbb S_3:G_2] = 3 \not = 2[/tex].
     
  4. Feb 13, 2009 #3
    Thank you so much!
     
  5. Feb 15, 2009 #4
    You are WELCOME!

    Well I also noticed I made a 'print' mistake...

    in the last row

    I wrote [tex][F:E][/tex] instead of
    [tex][F: F \cap E][/tex]

    but I guess you noticed the mistake and you got the right meaning.

    See you next time!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Please help! Field extension problem
  1. Field extensions (Replies: 5)

Loading...