We show deterministic polynomial time algorithms over some family of finite fields for solving univariate polynomial equations and some related problems such as taking nth roots, constructing nth nonresidues, constructing primitive elements and computing elliptic curve “nth roots”. In additional, we present a deterministic polynomial time primality test for some family of integers. All algorithms can be proved by elementary means (without assuming any unproven hypothesis). The problem of solving polynomial equations over finite fields is a generalization of the following problems over finite fields • constructing primitive nth roots of unity, • taking nth roots, • constructing nth nonresidues, • constructing primitive elements (generators of the multiplicative group) for any positive n dividing the number of elements of the underlying field. By the Tonelli-Shanks square root algorithm [21, 19] and its generalization for taking nth roots, constructing nth nonresidues