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Factorization in Krull monoids with infinite class group

Florian Kainrath (1999)

Colloquium Mathematicae

Let H be a Krull monoid with infinite class group and such that each divisor class of H contains a prime divisor. We show that for each finite set L of integers ≥2 there exists some h ∈ H such that the following are equivalent: (i) h has a representation h = u 1 · . . . · u k for some irreducible elements u i , (ii) k ∈ L.

Fundamental units for orders of unit rank 1 and generated by a unit

Stéphane R. Louboutin (2016)

Banach Center Publications

Let ε be an algebraic unit for which the rank of the group of units of the order ℤ[ε] is equal to 1. Assume that ε is not a complex root of unity. It is natural to wonder whether ε is a fundamental unit of this order. It turns out that the answer is in general yes, and that a fundamental unit of this order can be explicitly given (as an explicit polynomial in ε) in the rare cases when the answer is no. This paper is a self-contained exposition of the solution to this problem, solution which was...

Fundamental units in a family of cubic fields

Veikko Ennola (2004)

Journal de Théorie des Nombres de Bordeaux

Let 𝒪 be the maximal order of the cubic field generated by a zero ε of x 3 + ( ℓ - 1 ) x 2 - ℓ x - 1 for ℓ ∈ ℤ , ℓ ≥ 3 . We prove that ε , ε - 1 is a fundamental pair of units for 𝒪 , if [ 𝒪 : ℤ [ ε ] ] ≤ ℓ / 3 .

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