Results 1 to 4 of 4

Math Help - Maximal Ideals

  1. #1
    Junior Member
    Joined
    Feb 2010
    Posts
    42

    Maximal Ideals

    Prove <x^2+1> is maximal in R[x]

    R[x] = {all polynomials with real coefficients}

    <x^2+1> = {f(x)*(x^2+1) ; f(x) is from R[x]}

    Could you guys give me some hints? I am not really sure how to even start this. This exact problem is actually an example in my book but I could not even follow it! (Gallian)

    Thank you
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor FernandoRevilla's Avatar
    Joined
    Nov 2010
    From
    Madrid, Spain
    Posts
    2,162
    Thanks
    45
    One way (there is an immediate characterization in terms of irredutible polynomials):

    <x^2+1>\;\textrm{maximal}\;\Leftrightarrow \mathbb{R}[x]/<x^2+1>\;\textrm{is\:a\;field}

    and if I\subset \mathbb{R}[x] is an ideal, then \mathbb{R}/I is a commutative ring.

    So, you only have to prove that

    ax+b+<x^2+1>\;\;(ax+b\neq 0)

    is a unit in  \mathbb{R}[x]/<x^2+1> .


    Fernando Revilla
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Feb 2010
    Posts
    42
    Thanks I was actually trying to prove that R[x]/<x^2+1> was a field by showing <x^2+1> was maximal but I guess showing that its all the elements are units would be easier
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Jan 2011
    Posts
    2
    yes that is the easiest way to show that something is maximal. Think about what happens when you take the quotient R[x]/(x^2+1).

    Remember when you mod out by an ideal you partion the ring into cosets: a+(x^2+1) for each a in R[x]. Something gets "squashed" to 0 iff it is in the ideal (x^2+1).

    So what type of elements are left? What can you decide this quotient ring is isomorphic to? Think about this for a bit, you should be able to show it's a field without too much trouble.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Prime Ideals, Maximal Ideals
    Posted in the Advanced Algebra Forum
    Replies: 8
    Last Post: March 7th 2011, 08:02 AM
  2. Maximal Ideals
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: November 7th 2009, 09:17 AM
  3. Maximal ideals of C[x,y]
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: February 10th 2009, 04:49 PM
  4. maximal ideals
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: November 13th 2008, 08:31 PM
  5. Maximal Ideals of Q[x]
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: September 28th 2008, 07:26 PM

Search Tags


/mathhelpforum @mathhelpforum