Determining the fundamental unit of a pure cubic field given any unit

Abstract:

A number of algorithms which have been used to derive fundamental units for pure cubic fields suffer from the lack of absolute certainty that the units obtained are fundamental. We present here an algorithm which will correct this deficiency. Briefly, if $\eta$ is any nontrivial unit of a pure cubic field, then for some positive integer $N, {\eta ^{1/N}}$ will be a fundamental unit. Our method determines which of the real numbers ${\eta ^{1/N}}$ are integers of the field and, subsequently, will determine the coefficients of the fundamental unit. We illustrate the process with several numerical examples.