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Homework Help: Degree of a splitting field

  1. Nov 29, 2009 #1
    1. The problem statement, all variables and given/known data
    Let f in K[x] be a polynomial over a field K. De fine the notion of a splitting field
    L of f over K. Show that if deg f = d, then f has a splitting fi eld over K of degree
    dividing d!

    3. The attempt at a solution
    If f is reducible, then this seems true by induction. I'm not sure about the case where f is irreducible over K though.

    Thanks for your help.
  2. jcsd
  3. Nov 30, 2009 #2
    For anyone keeping score at home, I think the irreducible case is also by induction. Consider an intermediate field made by adjoining a root r of f to K, and then consider L as a splitting field over K(r) of g(x) (where f(x) = (x - r)g(x).)
    Last edited: Nov 30, 2009
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