Parameterization of nice polynomials

A univariable polynomial p(x) is said to be nice if all of its coefficients as well as all of the roots of both p(x) and its derivative p0(x) are integers. p(x) is called Q-nice polynomial if the coefficients, roots, and critical points are rational numbers. This research concentrates on finding parameterized families of symmetric polynomial with four, five, and seven roots. The relations between the roots and critical points of polynomials with four, five, and seven roots are considered respectively. By using the technique of parameterization and substitution, the pattern of solutions of the polynomials in the field of integer, rational, and Q(px) are observed. Then, based on the pattern of solutions, theorems will be constructed. Parameterized families of symmetric polynomials with four and five roots in the field of integral and rational numbers are obtained. Meanwhile, the roots and critical points for symmetric polynomials with seven roots are studied in the field of Q(px). Hence, parameterized families of symmetric polynomials with seven roots are found.

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