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Extension fields

  • Thread starter ehrenfest
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  • #1
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[SOLVED] extension fields

Homework Statement


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

Homework Equations





The Attempt at a Solution

 
Last edited:

Answers and Replies

  • #2
StatusX
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Do you mean algebraic over F? And what is irr([itex]\alpha[/itex],F)?
 
  • #3
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Yes. See the EDIT. irr(\alpha,F) is just the monic irreducible polynomial in F[x] that alpha is a zero of.
 
  • #4
morphism
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What are your thoughts about this? Have you tried to come up with a counterexample?
 
  • #5
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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.
 
  • #6
morphism
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But (x^2+2)(x^2-2) is not irreducible over Q.
 
  • #7
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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:
  • #8
morphism
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That's better!
 

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