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Math Help - Abstract Algebra: a problem about ideals

  1. #1
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    Abstract Algebra: a problem about ideals

    Let J be the set of all polynomials with zero constant term in Z[x].
    a.) Show that J is the principal ideal (x) in Z[x].
    b.) Show that Z[x]/J consists of an infinite number of distinct cosets, one for each nZ.
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  2. #2
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    Quote Originally Posted by iwonde View Post
    Let J be the set of all polynomials with zero constant term in Z[x].
    a.) Show that J is the principal ideal (x) in Z[x].
    b.) Show that Z[x]/J consists of an infinite number of distinct cosets, one for each nZ.
    I can't see the whole question because there is a "physicsforums.com" tag that hides some of your question.

    (a) is simply to verify that J is an ideal in Z[x] (Note that Z[x] is not a P.I.D. )
    (b) Z[x] / (x) \cong Z. You can find a surgective group homomorphism from Z[x] to Z whose kernel is (x). Then by the first isomorphism theorem, Z[x] / (x) \cong Z. It means Z[x]/J has an infinite number of distinct cosets.
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