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Principle Ideals of a Polynomial Quotient Ring

  1. Oct 15, 2015 #1
    1. The problem statement, all variables and given/known data

    Let A be the algebra [itex]\mathbb{Z}_5[x]/I[/itex] where [itex]I[/itex] is the principle ideal generated by [itex]x^2+4[/itex] and [itex]\mathbb{Z}_5[x][/itex] is the ring of polynomials modulo 5.

    Find all the ideals of A
    Let G be the group of invertible elements in A. Find the subgroups of the prime decomposition.


    2. Relevant equations
    None

    3. The attempt at a solution

    I have no idea where to start. Why is [itex]x^2+4[/itex] an ideal? How do I find other ideals?

    I have been asked about invertible elements in rings like [itex]\mathbb{Z}/n\mathbb{Z}[/itex] (just the elements co-prime to n) but how does this concepts relate to polynomials? Are invertible elements in polynomial rings also "coprime" in some sense??

    Thankyou
     
    Last edited: Oct 15, 2015
  2. jcsd
  3. Oct 16, 2015 #2

    andrewkirk

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    It's not, and the problem didn't say it was. It is the generator of a (principal) ideal.

    You won't be able to even get started on this if you don't know what an ideal is and what a principal ideal is. Your text and/or notes will have given you definitions.

    What are they?
     
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