ieja

International Electronic Journal of Algebra

1306-6048

Abdullah HARMANCI

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

Groves

Jonathan

12 01 2007

2 2 208 225 12 01 2007

2007

International Electronic Journal of Algebra

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.

Critical point D-nice polynomial integral domain nice polynomial polynomial root symmetric polynomial

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Department of Mathematics Patterson Oﬃce Tower 713 University of Kentucky Lexington, KY 40506-0027

E-mail: JGroves@ms.uky.edu, Jonny77889@yahoo.com