In this paper the author states
a class of infinitely many real cubic fields for which the Jacobi-Perron algorithm of a
properly chosen vector becomes periodic and calculates explicitly a fundamental unit
for each field. The main results of this paper are: Let m = a6+ 3a3+ 3,
ω =, m cube free a ∈ N; then the Jacobi-Perron algorithm of a(0)= (ω,ω2)
is periodic. The length of the primitive preperiod is four and the length
of the primitive period is three. A fundamental unit in Q(ω) is given by
e = a3+ 1 − aω.
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