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Extension fields

  1. Mar 1, 2008 #1
    [SOLVED] extension fields

    1. The problem statement, all variables and given/known data
    Let F be a field and let E be an extension field of F. Let [itex]\alpha[/itex] be an element of E that is algebraic over E. Is it true that all of the zeros of [itex]irr(\alpha,F)[/itex] are contained in the extension field [itex]F(\alpha)[/itex]?
    EDIT: I mean algebraic over F

    2. Relevant equations



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

    StatusX

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    Do you mean algebraic over F? And what is irr([itex]\alpha[/itex],F)?
     
  4. Mar 1, 2008 #3
    Yes. See the EDIT. irr(\alpha,F) is just the monic irreducible polynomial in F[x] that alpha is a zero of.
     
  5. Mar 1, 2008 #4

    morphism

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    What are your thoughts about this? Have you tried to come up with a counterexample?
     
  6. Mar 2, 2008 #5
    It is false. Take (x^2+2)(x^2-2) over the rationals. Then zeros of the first factor are imaginary and the zeros of the second factor are real.
     
  7. Mar 2, 2008 #6

    morphism

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    But (x^2+2)(x^2-2) is not irreducible over Q.
     
  8. Mar 2, 2008 #7
    Take x^3+3. It is irreducible by Eisenstein with p = 3. Its zeros are: [itex]\sqrt[3]{3}e^{ik\pi/3}[/itex] where k=0,1,2. Two of those values of k produce imaginary roots.
     
    Last edited: Mar 2, 2008
  9. Mar 2, 2008 #8

    morphism

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    That's better!
     
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