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Distribution of units of a cubic abelian field modulo prime numbers

We studied the distribution of units of an algebraic number field modulo prime ideals. Here we study the distribution of units of a cubic abelian field modulo rational prime numbers. For a decomposable prime number p, 2(p i 1) 2 is an upper bound of the order of the unit group modulo p, and we show that the conjectural density of primes which attain it is really positive.