SOLUBILITY TO THE LEGENDRE EQUATION IN THE POLYNOMIAL RING SOLUBILITY TO THE LEGENDRE EQUATION IN THE POLYNOMIAL RING \Phi_q[t]

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.