order reversing bijection - Fundamental theorem of Galois

Discussion in 'Undergraduate Math' started by nancy, Apr 12, 2007.

  1. nancy

    nancy Guest

    one of the parts of the galois theory says that
    the function Gamma:Sub(G) --> Lat(E/F), defined by H--> E^H, is an
    order reversing bijection with inverse delta: B --> Gal(E/B).

    Gamma(H):=E^H :={x is an element of E | for all h in H, h(x)=x|
    delta(k):=Gal(E/K)

    i know how to prove the bijection part of this theorem, but how can i show
    that this is order reversing?
    i knw that we need to show that
    if K1 is contained in K2 then delta of K2 is contained in delta of K1
    and if H1 is contained in H2 then gamma of H2 is contained in gamma of H1, but how?
    thanks alot
     
    nancy, Apr 12, 2007
    #1
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  2. Isn't it trivial? Suppose H is contained in K. Then E^K is the set of
    all elements of E that are fixed by every element of K, which includes
    every element of H. So every element of E^K is fixed by every element
    of H, hence E^K is contained in E^H.

    Conversely, if K is contained in F, then Gal(E/K) consists of all
    elements of G that fix K pointwise, while Gal(E/F) of all elements of
    G that fix F pointwise. If K is contained in F, then any element of
    Gal(E/F) will necessarily fix K, hence...

    --
    ======================================================================
    "It's not denial. I'm just very selective about
    what I accept as reality."
    --- Calvin ("Calvin and Hobbes" by Bill Watterson)
    ======================================================================

    Arturo Magidin
    magidin-at-member-ams-org
     
    Arturo Magidin, Apr 12, 2007
    #2
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