Fundamentality of a cubic unit u for ℤ[u]

Consider a cubic unit u of positive discriminant. We present a computational proof of the fact that u is a fundamental unit of the order Z[u] in most cases and determine the exceptions. This extends a similar (but restrictive) result due to E. Thomas. Introduction Let f(X) := X + aX + bX ± 1 ∈ Z[X] be irreducible in Z[X] and with three (distinct) real roots. We think of the order R := Z[u], obtained by adjoining a root u of f(X), as a subring of the real numbers. It is well known that the group U(R) of positive units of R is a free Abelian group of rank 2 and the unit-group of R is {−1, 1} × U(R). By a fundamental unit of R we mean a unit of R whose absolute value is a member of some free basis of U(R). Since u is clearly a unit of R, it is natural to ask when u is a fundamental unit of R. In his investigation [7] of fundamental units of cubic orders using Berwick’s algorithm, E. Thomas has defined a useful numerical function of the roots of f(X) which is denoted by θ(a, b) in the present article. In (3.1) of [7] Thomas proved that if θ(a, b) > 2, then u is a fundamental unit of R; he also indicated the necessity of some such restriction by alluding to the case of (a, b) = (2n, n), where n ≥ 3 is an integer, in which u fails to be a fundamental unit of R and in fact θ(2n, n) < 2. This result of Thomas is the cornerstone and the starting point of our investigation. Without any loss, we restrict ourselves to the case where f(0) = 1 and a < b, throughout the article. Our main theorem is Theorem. Let (a, b) be an ordered pair of integers with a < b such that f(a,b)(X) := X 3 + aX + bX + 1 is irreducible in Z[X] and has three distinct real roots. Let u be a real number with f(a,b)(u) = 0 and let R := Z[u]. Then, u is a fundamental unit of R if and only if (a, b) = (2n, n) for any integer n ≥ 3 and (a, b) = (5, 6). Our proof is almost entirely computational in nature, involving symbolic as well as (real) numerical computation. In order to estimate the values of θ(a, b) for the integer pairs (a, b) of interest, we partition their natural domain into 15 parts. Then, Mathematica is harnessed to compute the real extrema of appropriate rational functions of two variables on each of the parts to determine whether θ(a, b) exceeds Received by the editor April 2