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Homework Help: Field theory problem

  1. Mar 1, 2008 #1
    [SOLVED] field theory problem

    1. The problem statement, all variables and given/known data
    If F is a field that has characteristic p, it must contain a copy of Z_p. Is it true that F must sit inside of the algebraic closure of Z_p? My book assumes that it does and I do not understand why?


    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Mar 1, 2008 #2

    morphism

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    Are you sure it says F is an arbitrary field of characteristic p? Because this is obviously false. For instance take the rational function field of the alg closure of Z_p - this still has characteristic p and properly contains the alg closure of Z_p.

    On the other hand, this is true if F is a finite field of characteristic p.
     
  4. Mar 1, 2008 #3
    Sorry. F has p^r elements.
     
  5. Mar 1, 2008 #4

    morphism

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    Then F is the splitting field of x^(p^r) - x over Z_p.
     
  6. Mar 1, 2008 #5
    I am in section 33 and splitting fields are in section 50. Thus, I do not even know what they are.

    I figured it out though. F must be a finite extension of Z_p and finite extensions are always algebraic extensions and thus F must sit inside the algebraic closure of Z_p.
     
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