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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!
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