1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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

    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??

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


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted