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Math Help - polynomial

  1. #1
    Senior Member Sampras's Avatar
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    polynomial

    Let  J be the set of all polynomials with zero constant term in  \mathbb{Z}[x] .

    (a) Show that  J is the principal ideal  (x) in  \mathbb{Z}[x] .

    (b) Show that  \mathbb{Z}[x]/J consists of an infinite number of distinct cosets.

    So for (a) suppose it is not? For (b) you would also use contradiction?
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  2. #2
    Junior Member
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    I would think about it a little more directly. First, J is an ideal because any polynomial multiplied by an element of J will give you an element of J. It is principal because it is generated by x.

    For the second part, I can't seem to give a rigourous explanation that I am happy with. Try to write a few of the cosets.
    \lbrace 1 + x, 2 + x, 3 + x,\ldots\rbrace
    \lbrace 1 + x^2, 2 + x^2, 3 + x^2,\ldots, x+x^2, 2x+x^2, 3x+x^2 \ldots \rbrace

    They are all different and there is an infinite number of them because you can just keep raising x to a higher power. I hope I didn't confuse you with that. I'll think about it some more and try to come up with a better explanation.
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