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Math Help - Borel field Algebra

  1. #1
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    Borel field Algebra

    prove that any F[x]/ <r(x)>, r(x) being irreducible is a PID
    Last edited by karlito03; April 2nd 2011 at 02:20 AM. Reason: Mistyped, the wrong question
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    " alt="\mathbb{F}/(\;r(x)\" /> is a field so, it has only two ideals:


    \quad (ii)\quad\mathbb{F}/(\;r(x)\=(\;1+(r(x))\" alt="(i)\quad\{0+(r(x))\}=(\;0+(r(x))\\quad (ii)\quad\mathbb{F}/(\;r(x)\=(\;1+(r(x))\" /> .
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  3. #3
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    and any field is a PID. perhaps what you meant to ask is: prove that if F is a field, and thus F[x] is a PID, then F[x]/<r(x)> is a field when r(x) is irreducible.

    the proof might go something like r(x) irreducible --> <r(x)> is maximal, so F[x]/<r(x)> has no proper ideals.
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