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- Title
The Frobenius problem over number fields with a real embedding.
- Authors
Feiner, Alex; Hefty, Zion
- Abstract
Given a number field K with at least one real embedding, we generalize the notion of the classical Frobenius problem to the ring of integers O K of K by describing certain Frobenius semigroups, Frob (α 1 , ... , α n) , for appropriate elements α 1 , ... , α n ∈ O K . We construct a partial ordering on Frob (α 1 , ... , α n) , and show that this set is completely described by the maximal elements with respect to this ordering. We also show that Frob (α 1 , ... , α n) will always have finitely many such maximal elements, but in general, the number of maximal elements can grow without bound as n is fixed and α 1 , ... , α n ∈ O K vary. Explicit examples of the Frobenius semigroups are also calculated for certain cases in real quadratic number fields.
- Subjects
RINGS of integers; QUADRATIC fields; ALGEBRAIC numbers; ALGEBRAIC fields
- Publication
Semigroup Forum, 2024, Vol 108, Issue 1, p73
- ISSN
0037-1912
- Publication type
Article
- DOI
10.1007/s00233-023-10403-9