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Math Help - about finite fields

  1. #1
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    about finite fields

    Let K be a finite feld. Show that for any x\in K there is an f\in K[X_{1},...,X_{n}] with f(x)=1 and f(y)=0 for all y \in K\\{x}.

    Thanks in advance.
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  2. #2
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    Quote Originally Posted by Biscaim View Post
    Let K be a finite feld. Show that for any x\in K there is an f\in K[X_{1},...,X_{n}] with f(x)=1 and f(y)=0 for all y \in K\\{x}.

    Thanks in advance.
    Your question makes no sense. You cannot evaluate f as a single point in K if n>1. Thus, I am going to show that for any x\in K there is f\in K[X] with f(x) = 1 and f(y) = 0 for all y\in K - \{x\}. Since K is a finite field we can define g(X) = \Pi_{a\in K-\{x\}} (X-a). We see that g(x) \not = 0 and g(y) = 0 for all y\in K-\{x\}. Define f(X) = \tfrac{1}{g(x)}g(X).
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  3. #3
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    about finite fields

    Sorry for the typo. The correct question was.

    Let K be a finite feld. Show that for any x\in K^{n} there is an f\in K[X_{1},...,X_{n}] with f(x)=1 and f(y)=0 for all y \in K^{n}-\{x\}.

    Thanks again.
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  4. #4
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    Quote Originally Posted by Biscaim View Post
    Sorry for the typo. The correct question was.

    Let K be a finite feld. Show that for any x\in K^{n} there is an f\in K[X_{1},...,X_{n}] with f(x)=1 and f(y)=0 for all y \in K^{n}-\{x\}.

    Thanks again.
    It is the same idea! Try to modify what I first wrote to K^n.
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