Polynomials of degree 4 defining units
Osnel BrocheUniversidade Federal de Lavras, Brazil
Ángel del RíoUniversidad de Murcia, Spain
If is the generator of a cyclic group of order then every element of the group ring is the result of evaluating at a polynomial of degree smaller than with integral coefficients. When such an evaluation result into a unit we say that the polynomial defines a unit on order . Marciniak and Sehgal have classified the polynomials of degree at most 3 defining units. The number of such polynomials is finite. However the number of polynomials of degree 4 defining units on order 5 is infinite and we give the full list of such polynomials. We prove that (up to a sign) every irreducible polynomial of degree 4 defining a unit on an order greater than 5 is of the form and obtain conditions for a polynomial of this form to define a unit. As an application we prove that if is greater than 5 then the number of polynomials of degree 4 defining units on order is finite and for we give explicitly all the polynomials of degree 4 defining units on order . We also include a conjecture on what we expect to be the full list of polynomials of degree 4 defining units, which is based on computer aided calculations.
Cite this article
Osnel Broche, Ángel del Río, Polynomials of degree 4 defining units. Rev. Mat. Iberoam. 33 (2017), no. 4, pp. 1487–1499DOI 10.4171/RMI/979