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Thread: Mappings and Polynomials

  1. #1
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    Mappings and Polynomials

    Show that for every mapping $\displaystyle g:\mathbb{Z}/p\mathbb{Z}\rightarrow\mathbb{Z}/p\mathbb{Z}$, that there exists a polynomial $\displaystyle f(x)\in\mathbb{Z}/p\mathbb{Z}[x]$ such that $\displaystyle f(a) = g(a) $ for all $\displaystyle a\in\mathbb{Z}/p\mathbb{Z}$

    I'm pretty sure Lagrange Interpolation is what is needed here but I'm not sure how to use it.
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  2. #2
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    Yes since the sets used are finite with the same cardinal and that $\displaystyle \mathbb{Z}_p-\{0\}$ is a group for $\displaystyle \times$ (I assume $\displaystyle p$ denotes a prime) you can write something like

    $\displaystyle f:x\mapsto\sum\limits_{a\in \mathbb{Z}_p}\prod\limits_{k\in\mathbb{Z}_p-\{a\}}\frac{(x-k)g(a)}{(a-k)}$ , and it is an element of $\displaystyle \mathbb{Z}_p[x]$
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