Polynomials of degree 4 defining units

  • Osnel Broche

    Universidade Federal de Lavras, Brazil
  • Ángel del Río

    Universidad de Murcia, Spain

Abstract

If xx is the generator of a cyclic group of order nn then every element of the group ring Zx\mathbb Z \langle x \rangle is the result of evaluating xx at a polynomial of degree smaller than nn with integral coefficients. When such an evaluation result into a unit we say that the polynomial defines a unit on order nn. 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 a(X4+1)+b(X3+X)+(12a2b)X2a(X^4+1)+b(X^3+X)+(1-2a-2b)X^2 and obtain conditions for a polynomial of this form to define a unit. As an application we prove that if nn is greater than 5 then the number of polynomials of degree 4 defining units on order nn is finite and for n10n\le 10 we give explicitly all the polynomials of degree 4 defining units on order nn. 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. 1487–1499

DOI 10.4171/RMI/979