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Math Help - maximal ideal question

  1. #1
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    maximal ideal question

    Conser Z[x] = {all polynomials with integer coefficients} and I={f(x) in Z[x] such that f(0) is even} Note: I = <x,2>

    I have already shown that I = <x,2>

    Now I must show whether I is prime or maximal or neither and also how many elements Z[x]/I has

    I know that I is maximal and that Z[x]/I but I cannot for the life of me understand why it has only 2 elements (odds and evens I'm guessing) I don't think i fully understand Factor rings

    So Z[x]/I = {f(x) + <x,2>} which can be rewritten as {x*g(x) + k + <x,2>} and then <x,2> absorbs x*g(x)

    so Z[x]/I = {k + <x,2>}

    This is as far as I got. Can anyone help?
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  2. #2
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    Quote Originally Posted by mulaosmanovicben View Post
    Conser Z[x] = {all polynomials with integer coefficients} and I={f(x) in Z[x] such that f(0) is even} Note: I = <x,2>

    I have already shown that I = <x,2>

    Now I must show whether I is prime or maximal or neither and also how many elements Z[x]/I has

    I know that I is maximal and that Z[x]/I but I cannot for the life of me understand why it has only 2 elements (odds and evens I'm guessing) I don't think i fully understand Factor rings

    So Z[x]/I = {f(x) + <x,2>} which can be rewritten as {x*g(x) + k + <x,2>} and then <x,2> absorbs x*g(x)

    so Z[x]/I = {k + <x,2>}

    This is as far as I got. Can anyone help?


    Any element in \mathbb{Z}[x] has free coefficient either even or odd, so awny element modulo the ideal I is

    either 0 or 1...

    Tonio
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  3. #3
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    [xCan someone please explain this more? I do not see why Z[x]/I has only 2 elements!
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