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Homework Help: Field of characteristic 0

  1. Nov 12, 2008 #1
    1. The problem statement, all variables and given/known data

    Show that any field of characteristic 0 is perfect.

    2. The attempt at a solution

    Let F be a field of characteristic 0.
    Let K be a finite extension of F.
    Let b be an element in K .

    I need to show that b satisfies a polynomial over F having no multiple roots.

    If f(x) is irreducible in F[x] then f(x) has no multiple roots.

    I need to show that b satisfies a irreducible polynomial in F[x].

    Well, suppose b can't satisfy any irreducible polynomial in F[x]. Can I get a contradiction? What kind of element could I have that didn't satisfy any irreducible polynomial?

    Then how can b be in the finite extension...? A finite extension for a field of characteristic 0 is of the form F(a), it is generated by a single element.


    I'm stuck. I don't even know if what I've laid out so far is correct.

    I'm having trouble connecting the arbitrary element b to a polynomial-- It's not obvious to me that b is the root of any polynomial in F[x].
     
    Last edited: Nov 12, 2008
  2. jcsd
  3. Nov 14, 2008 #2

    morphism

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    Science Advisor
    Homework Helper

    Well K is a finite extension of F. So b is algebraic over F, and hence has a minimal polynomial in F[x].

    If you don't know what a min poly is, think about what it means for K to be a finite extension of F. What does this say about the set {1, b, b^2, b^3, ...}?
     
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