Analysis determines nontrivial solutions to the Legendre equation in polynomial rings over finite fields, suggesting a method for construction.
We determine the necessary and sufficient conditions for the existence of nontrivial solutions to the Legendre equation over the ring of polynomials , where is a finite field. Additionally, we present a method for determining whether the Legendre equation has nontrivial solutions in . When such a solution exists, we can construct an infinite number of solutions in via the Réalis formulae.
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Diaz-Vargas et al. (2025) studied this question.
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