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Math Help - Ideals in a UFD and PID

  1. #1
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    Ideals in a UFD and PID

    This problem is from Dummit and Foote Abstract Algebra

    Exhibit all of the ideals in the ring

    F[x]/(p(x)) , where F is a field and p(x) is a polynomial in F[x] (describe them in terms of the factorization of p(x)))

    I know that since F is a field that F[x] is a PID and a UFD.

    So since p(x) \in F[x] p(x)=p_1(x)p_2(x) \cdots p_n(x) and where each p_i(x) is irreduceable and this is unique upto associates. Also since each p_i(x)|p(x) the Ideal of (p(x)) \subseteq p_i(x)

    First I am not sure if the above observations help me, If they do I don't see where to go from here. Second I am having a hard time visualzing what elements of this ring would look like. I think it would eliminate all elements of F[x] that have the same degree as any factor of P_i(x).

    Thanks

    TES
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  2. #2
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    Quote Originally Posted by TheEmptySet View Post
    This problem is from Dummit and Foote Abstract Algebra

    Exhibit all of the ideals in the ring

    F[x]/(p(x)) , where F is a field and p(x) is a polynomial in F[x] (describe them in terms of the factorization of p(x)))

    I know that since F is a field that F[x] is a PID and a UFD.

    So since p(x) \in F[x] p(x)=p_1(x)p_2(x) \cdots p_n(x) and where each p_i(x) is irreduceable and this is unique upto associates. Also since each p_i(x)|p(x) the Ideal of (p(x)) \subseteq p_i(x)

    First I am not sure if the above observations help me, If they do I don't see where to go from here. Second I am having a hard time visualzing what elements of this ring would look like. I think it would eliminate all elements of F[x] that have the same degree as any factor of P_i(x).

    Thanks

    TES
    suppose p(x)=\prod_{j=1}^n (p_j(x))^{r_j}, \ r_j \geq 1, be the factorization of p(x) into irreducibles. an ideal of F[x]/(p(x)) is in the form I/(p(x)), where I is an ideal of F[x] which contains (p(x)).

    we have I=(f(x)), for some f(x) \in F[x], since F[x] is a PID. now (p(x)) \subseteq I is equivalent to f(x) \mid p(x), i.e. f(x)=\prod_{j=1}^n (p_j(x))^{s_j}, \ 0 \leq s_j \leq r_j.
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  3. #3
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    Thank you very much .
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