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Linear Map = Function of degree P-1

  1. Oct 26, 2009 #1
    If p is prime, prove that for every function f: Fp -> Fp there exists a polynomial Q (depending on f) of degree at most p-1 such that f(x) = Q(x) for each x in Fp.
     
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  3. Oct 29, 2009 #2
    Would polynomial interpolation work here?
     
  4. Oct 29, 2009 #3

    HallsofIvy

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    I don't see how you could interpolate. There are only p possible values in [itex]F_p[/itex]. I notice you titled this "Linear Map= Function of degree P-1". Do you understand that there is nothing said here about f being linear?
     
  5. Oct 29, 2009 #4
    Yea I assumed by accident it was linear. Yea I understand that there are only p values in Fp but I don't know how to make the connection to a polynomial. I mean I know a polynomial of degree p-1 has p coefficients but for some reason I can't connect the dots.
     
  6. Oct 29, 2009 #5

    Hurkyl

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    Have you tried it? What happened?


    Polynomial interpolation works over any field... or is there something else you see that I don't?
     
  7. Oct 30, 2009 #6

    HallsofIvy

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    Well, I think of "interpolation" as finding values between given values. And since this is a finite field, there is nothing "between" values.
     
  8. Oct 30, 2009 #7

    Hurkyl

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