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Math Help - Long division and extension fields

  1. #1
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    Long division and extension fields

    I need some help understanding the following worked example:



    So they have \alpha = x +I where I = <p(x)>. But what do they mean "by long division we have..."? Did they divide p(x) by \alpha? If so how could they end up with (x-\alpha) (x+2+\alpha)? Any help is really appreciated.
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  2. #2
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    Quote Originally Posted by demode View Post
    I need some help understanding the following worked example:



    So they have \alpha = x +I where I = <p(x)>. But what do they mean "by long division we have..."? Did they divide p(x) by \alpha? If so how could they end up with (x-\alpha) (x+2+\alpha)? Any help is really appreciated.


    You know (or should to) that \alpha:= x+I is a root of p(x)\Longrightarrow (x-\alpha)\mid p(x) , and thus you can

    (long) divide these two polynomials.

    Note that the remainder of this division is \alpha^2+2\alpha+3=0 , as expected.

    Tonio
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