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Thread: The Maximal ideals of R[x]

  1. #1
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    The Maximal ideals of R[x]

    So I am in the process of trying to do a proof, and I need show that the maximal ideals in \mathbb{R}[x] are the irreducible polynomials of the form



    What I am confused about is: why can't a polynomial with degree >2, generate a maximal ideal in \mathbb{R}[x]?
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by CropDuster View Post
    What I am confused about is: why can't a polynomial with degree >2, generate a maximal ideal in \mathbb{R}[x]?

    For example, if p ( x ) in IR [ x ] has degree 3 then,

    p ( x ) = ( x - a ) q ( x ) with q ( x ) in IR [x] and a in IR .

    Then,

    ( p( x ) ) is contained in ( x - a ) and ( x - a ) is different from ( p ( x ) ) and IR [ x ] .


    Edited: I forgot to say that p ( x ) had degree 3 .
    Last edited by FernandoRevilla; Apr 24th 2011 at 11:24 AM.
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  3. #3
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    Thanks. I think I see what you're saying, but what does IR[x] denote? And, p(x) is irreducible in IR[x] right? And if so:



    right?
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by CropDuster View Post
    Thanks. I think I see what you're saying, but what does IR[x] denote?

    Denotes




    And, p(x) is irreducible in IR[x] right? A:

    I proposed a polynomial of degree 3. It is easy to generalize.
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  5. #5
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    Quote Originally Posted by FernandoRevilla View Post
    For example, if p ( x ) in IR [ x ] has degree 3 then,

    p ( x ) = ( x - a ) q ( x ) with q ( x ) in IR [x] and a in IR .

    Then,

    ( p( x ) ) is contained in ( x - a ) and ( x - a ) is different from ( p ( x ) ) and IR [ x ] .


    Edited: I forgot to say that p ( x ) had degree 3 .
    So what if p(x) is irreducible and has degree 2?
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  6. #6
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by CropDuster View Post
    So what if p(x) is irreducible and has degree 2?

    Use that in a principal ideal domain every nonzero prime ideal is maximal.
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