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Ring of integers
About: Ring of integers is a research topic. Over the lifetime, 1856 publications have been published within this topic receiving 15882 citations.
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TL;DR: In this paper, a torsion-free subgroup acting discontinuously on 3-dimensional hyperbolic space is shown to be homeomorphic to the complement of a link in S3.
Abstract: Let be a torsion-free subgroup acting discontinuously on 3-dimensional hyperbolic space . Assume further that Γ\ℍ3 has finite hyperbolic volume. The quotient-space Γ\ℍ3 is then a 3-manifold which can be compactified by the addition of finitely many 2-tori. This paper discusses a procedure which decides whether Γ\ℍ3 is homeomorphic to the complement of a link in S3. We apply our procedure to subgroups of low index in , where is the ring of integers in . As a result we find new link complements having a complete hyperbolic structure coming from an arithmetic group. Finally we prove that up to conjugacy there are only finitely many commensurability classes of arithmetic subgroups so that Γ\ℍ3 is homeomorphic to the complement of a link in S3.
24 citations
TL;DR: Besancon et al. as discussed by the authors constructed new families of infinitely many fields K with non-monogenic phenomena, which supplement (J. Math. Fac. Sci Besancon, Theor. Nombres (1984) 25pp) and with monogenic (3).
24 citations
TL;DR: In this paper, it was shown that the ring of all endomorphisms of J(Cf, p) coincides with a ring of integers in the pth cyclotomic field.
Abstract: Let K be a field of characteristic zero, n ≥ 5 an integer, f(x) an irreducible polynomial over K of degree n, whose Galois group contains a doubly transitive simple non-abelian group. Let p be an odd prime, \({\mathbb{Z}[\zeta_p]}\) the ring of integers in the pth cyclotomic field, Cf, p : yp = f(x) the corresponding superelliptic curve and J(Cf, p) its jacobian. Assuming that either n = p + 1 or p does not divide n(n − 1), we prove that the ring of all endomorphisms of J(Cf, p) coincides with \({\mathbb{Z}[\zeta_p]}\) . The same is true if n = 4, the Galois group of f(x) is the full symmetric group S4 and K contains a primitive pth root of unity.
24 citations
TL;DR: In this paper, the maximal order of purely cubic fields such that their maximal order is determined by their units is determined, i.e., the order in which the fields have maximal order.
Abstract: All purely cubic fields such that their maximal order is generated by its units are determined.
24 citations
TL;DR: Chinburg and Friedman as mentioned in this paper showed that there exists an embedding σ:L→A over k which gives an optimal embedding of B into E in the sense that σ(L)∩E=σ(B).
23 citations