D-NICE SYMMETRIC POLYNOMIALS WITH FOUR ROOTS OVER INTEGRAL DOMAINS D OF ANY CHARACTERISTIC

Abstract

Let D be any integral domain of any characteristic. A polynomial p(x) ∈ D[x] is D-nice if p(x) and its derivative p′(x) split in D[x]. We give a complete description of all D-nice symmetric polynomials with four roots over integral domains D of any characteristic not equal to 2 by giving an explicit formula for constructing these polynomials and by counting equivalence classes of such D-nice polynomials. To illustrate our results, we give several examples we have found using our formula. We conclude by stating the open problem of finding all D-nice symmetric polynomials with four roots over integral domains D of characteristic 2 and all D-nice polynomials with four roots over all integral domains D of any characteristic.