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Math Help - checking answer on units of integer

  1. #1
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    checking answer on units of integer

    U(Z_4) ={ 1, 3}. Z_4 is integers mod n
    In general, U(Z_n) = { a in Z_n: hcf(a,n) =1}
    And
    U(Z_4[X]) = { 0, 1, 2 ,3}. Is this correct?
    where Z_n[X] is polynomial ring over Z generated by one variable
    How do you generate the formula of U(Z_n[X])?

    Thank you
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  2. #2
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    Quote Originally Posted by knguyen2005 View Post
    U(Z_4) ={ 1, 3}. Z_4 is integers mod n
    In general, U(Z_n) = { a in Z_n: hcf(a,n) =1}
    And
    U(Z_4[X]) = { 0, 1, 2 ,3}. Is this correct?


    Nop. In general, U(R[x])=U(R) , when R is a ring, so in this case the units of the polynomial ring are the same as the ring's.

    Tonio

    where Z_n[X] is polynomial ring over Z generated by one variable
    How do you generate the formula of U(Z_n[X])?

    Thank you
    .
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  3. #3
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    Thank you Tonio, but I think you misunderstood my question
    I know in general but what I meant was how do you write down the elements of the set
    Example: U(Z_4)={1, 3} since these elements of the set are coprime to 4
    Ok, how about the polynomial of the ring over Z_4. Can we find the elements in U(Z_4[x]) ?
    is it U(Z_4[X]) = { 0, 1, 2 ,3} or U(Z_4[X]) = { 1, 2 ,3} ?
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  4. #4
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    if R is a commutative ring with identity element, then U(R[x])=U(R) holds only for reduced rings, i.e. rings with no non-zero nilpotent elements.

    in general f(x)=a_0 + a_1x + \cdots + a_n x^n \in U(R[x]) if and only if a_0 \in U(R) and a_j is nilpotent for all j \geq 1. see my post in here: http://www.mathhelpforum.com/math-he...ial-rings.html

    for example if R=\mathbb{Z}/4\mathbb{Z}, then f(x)=a_0+a_1x + \cdots + a_nx^n \in U(R[x]) if and only if a_0 \in \{\bar{1}, \bar{3} \} and a_j \in \{\bar{0}, \bar{2} \}, \ \forall j \geq 1.
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