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 $\mathbb{Z}⟨x⟩$ 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 $a(X^4+1)+b(X^3+X)+(1-2a-2b)X^2$ and obtain conditions for a polynomial of this form to define a unit. As an application we prove that if $n$ is greater than 5 then the number of polynomials of degree 4 defining units on order $n$ is finite and for $n\le 10$ 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.