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Thread: Polynomials

  1. #1
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    Polynomials

    Let a(x) and b(x) be in F[x]. I need to prove that if a(x) and b(x) determine the same function, and if the number of elements in F exceeds the degree of a(x) as well as the degree of b(x), then a(x) = b(x).

    I know what if a(x) and b(x) determine the same function means. It means that a(x) - b(x) gives you the zero function. I dont know what if the number of elements in F exceeds the degree of a(x) as well as the degree of b(x) means. I also do not see how either of the two can conclude a(x) = b(x). Can anyone help me?
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  2. #2
    Junior Member Nehushtan's Avatar
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    Re: Polynomials

    The terminology is certainly confusing. I think what you mean by “zero function” is the constant function on $\displaystyle F$ whose image is 0, not the zero polynomial in $\displaystyle F[x]$. In other words, $\displaystyle a(x)$ and $\displaystyle b(x)$ determine the same function on F iff $\displaystyle a=b$ considered as functions $\displaystyle a,b:F\to F$.

    An example to show what I mean. Suppose $\displaystyle F=\mathbb F_2=\{0,1\}$, $\displaystyle a(x)=x$, $\displaystyle b(x)=x^2$. Then, considered as functions $\displaystyle F\to F$, $\displaystyle a$ and $\displaystyle b$ are equal (to the identity function on $\displaystyle F$), but $\displaystyle a(x)\ne b(x)$ as polynomials in $\displaystyle F[x]$.

    In the example, the number of elements of $\displaystyle F$ is not greater than the degree of $\displaystyle a(x)$. What you are asked to show is that if $\displaystyle |F|>\deg a,\deg b$, then $\displaystyle a(x)=b(x)$ as polynomials in $\displaystyle F[x]$ if $\displaystyle a=b$ as functions $\displaystyle F\to F$.
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