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A hint on a field extension problem?

  1. Apr 6, 2013 #1
    *** This is *not* homework. All I'd like is a push in the right direction.***

    1. The problem statement, all variables and given/known data
    Let [itex]K / F[/itex] be an algebraic field extension and [itex]R[/itex] a ring such that [itex]F \subset R \subset K[/itex]. Show that [itex]R[/itex] is a field.
    2. Relevant equations
    (none)
    3. The attempt at a solution
    I know that every element in [itex]R[/itex] is algebraic over [itex]F[/itex] and that every element in [itex]R \setminus{F}[/itex] must have an inverse in [itex]K\setminus{F}[/itex]. It suffices that these inverses are in [itex]R\setminus{F}[/itex], but I'm stuck trying to find a way to prove that this is the case. I've explored minimal polynomials of the inverse, but that doesn't seem to prove anything. Hrlp.
     
    Last edited: Apr 6, 2013
  2. jcsd
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