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.

Nonunique factorization and principalization in number fields

Abstract: Following what is basically Kummer’s relatively neglected approach to nonunique factorization, we determine the structure of the irreducible factorizations of an element n in the ring of integers of a number field K. Consequently, we give a combinatorial expression for the number of irreducible factorizations of n in the ring. When K is quadratic, we show in certain cases how quadratic forms can be used to explicitly produce all irreducible factorizations of n.

The Pell equation over the Fibonacci ∘-ring

Abstract: The paper considers the Pell equation $$N_1 \circ N_1 - A \circ N_2 \circ N_2 = 1$$ over the Fibonacci ∘-ring $$\mathop \mathbb{Z}\limits^ \circ $$ , which is obtained by supplying the ring of integers ℤ with the Fibonacci circle multiplication operation N ∘ M. It is proved that if a positive integer A satisfies the condition Aτ < [(A + 1)τ], where $$\tau = \tfrac{{ - 1 + \sqrt 5 }}{2}$$ is the golden section, and [x] is the integral part of x, then the Pell equation is solvable both in integers and in positive integers N1 and N2. Moreover, for the number n(A; X) of integer solutions (N1, N2), |N1| ≤ X, lower bounds are established. Bibliography: 7 titles.

Universally defining finitely generated subrings of global fields

Abstract: It is shown that any finitely generated subring of a global field has a universal first-order definition in its fraction field. This covers Koenigsmann's result for the ring of integers and its subsequent extensions to rings of integers in number fields and rings of $S$-integers in global function fields of odd characteristic. In this article a proof is presented which is uniform in all global fields, including the characteristic two case, where the result is entirely novel. Furthermore, the proposed method results in universal formulae requiring significantly fewer quantifiers than the formulae that can be derived through the previous approaches.

Power integral bases for real cyclotomic fields

Abstract: We consider the problem of determining all power integral bases for the maximal real subfield Q (ζ + ζ−1) of the p-th cyclotomic field Q (ζ), where p ≥ 5 is prime and ζ is a primitive p-th root of unity. The ring of integers is Z[ζ+ζ−1] so a power integral basis always exists, and there are further non-obvious generators for the ring. Specifically, we prove that if or one of the Galois conjugates of these five algebraic integers. Up to integer translation and multiplication by −1, there are no additional generators for p ≤ 11, and it is plausible that there are no additional generators for p > 13 as well. For p = 13 there is an additional generator, but we show that it does not generalise to an additional generator for 13 < p < 1000.

Accessing the Farrell-Tate cohomology of discrete groups

Abstract: We introduce a method to explicitly determine the Farrell-Tate cohomology of discrete groups. We apply this method to the Coxeter triangle and tetrahedral groups as well as to the Bianchi groups, i.e. PSL_2 over the ring of integers in an imaginary quadratic number field. We show that the Farrell-Tate cohomology of the Bianchi groups is completely determined by the numbers of conjugacy classes of finite subgroups. In fact, our access to Farrell-Tate cohomology allows us to detach the information about it from geometric models for the Bianchi groups and to express it only with the group structure. Formulae for the numbers of conjugacy classes of finite subgroups in the Bianchi groups have been determined in a thesis of Kramer, in terms of elementary number-theoretic information on the ring of integers. An evaluation of these formulae for a large number of Bianchi groups is provided numerically in the appendix. Our new insights about the homological torsion allow us to give a conceptual description of the cohomology ring structure of the Bianchi groups.