Polynomials of degree 4 defining units

Abstract

If x is the generator of a cyclic group of order n then every element of the group ring Zx is the result of evaluating x at a polynomial of degree smaller than n with integral coefficients. When such an evaluation result into a unit we say that the polynomial defines a unit on order n. 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) + (1 − 2a − 2b)X2 and obtain conditions for a polynomial of this form to define a unit. As an application we prove that if n is greater than 5 then the number of polynomials of degree 4 defining units on order n is finite and for n ≤ 10 we give explicitly all the polynomials of degree 4 defining units on order n. 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.