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For Chebyshev-polynomials on finite fields $F_p$ with a prime $p$, I have found the following expression:

$$ T_n(x)=\frac{(x+\sqrt{x^2-1})^n+(x-\sqrt{x^2-1})^n}{2} mod\space p. $$

But obviously, $\sqrt{x^2-1}$ is not defined on $F_p$ for some $x$. How can this formula still be a correct representation of Chebyshev-polynomials modulo $p$?

Thanks so much for some help!

Ada
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1 Answers1

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The simple fact is that the expression $$(x+\alpha)^n+(x-\alpha)^n$$ contains no odd powers of $\alpha$.

TonyK
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  • Thanks so much for responding! Do you know how to prove the expression above? When I set

    $a=x+\sqrt{x^2-1}$

    to use the formula

    $T_n(\frac{1}{2}(a+a^{-1}))=\frac{1}{2}(a^n+a^{-n})$

    I once again have the problem, that $a$ may not be defined for some $x$... Do you know an alternative there?

     – Ada Mar 15 '20 at 09:05
  • @Ada : If a square root is not in $\Bbb F_p$, then it is in $\Bbb F_{p^2}$. Thus compute in the quadratic extension and check whether the coefficients are actually in $\Bbb F_p$, i.e. the "imaginary" parts cancel out. You get the quadratic extension by adjoining a square root of a non-square. – emacs drives me nuts Apr 24 '20 at 22:03