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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, the authors give necessary and sufficient conditions under which a Galois extension of local fields of characteristic 0 is a cohomologically trivial group over the ring of integers in K. This has applications to elliptic curves over local fields and to ray classes of number fields.
Abstract: Let $L/K$ be a Galois extension of local fields of characteristic $0$ with Galois group $G$. If $\mathcal{F}$ is a formal group over the ring of integers in $K$, one can associate to $\mathcal F$ and each positive integer $n$ a $G$-module $F_L^n$ which as a set is the $n$-th power of the maximal ideal of the ring of integers in $L$. We give explicit necessary and sufficient conditions under which $F_L^n$ is a cohomologically trivial $G$-module. This has applications to elliptic curves over local fields and to ray class groups of number fields.
3 citations
TL;DR: In this paper, the second invariant of an extension of number fields defined by Chinburg via the canonical class of the extension and lying in the locally free class group was investigated, and it was shown that in Queyrut's S -class group, where S is a (finite) set of primes, the image of Chinburg's invariant equals the stable isomorphism of the ring of integers.
3 citations
TL;DR: This work investigates the special case when L is the ring of integers of the cyclotomic field of order m and S is the corresponding set of unit roots, and compute the coordinator polynomial explicitly when m = p and m = 2p, with p an odd prime.
Abstract: The coordinator polynomial of a lattice L is the numerator of its growth series as an abelian group, w.r.t, to a given set of generators S. We investigate the special case when L is the ring of integers of the cyclotomic field of order m and S is the corresponding set of unit roots. We compute it explicitly when m = p and m = 2p, with p an odd prime. This confirms, for small p, a conjecture of Parker. Our approach is geometric and is grounded in the theory of Ehrhart polynomials.
3 citations
TL;DR: The explicit computations for the ring $\mathbf{Z}_n$, the ring of integers modulo $n$, have been obtained and bounds of this probability of a finite commutative ring with identity 1 are found.
Abstract: In this paper, the probability that two elements of a finite ring have product zero is considered. The bounds of this probability are found for an arbitrary finite commutative ring with identity 1. An explicit formula for this probability in the case of, the ring of integers modulo, is obtained.
3 citations
TL;DR: In this paper, the authors studied discrete dynamical systems of the kind h(x) = x + g(x), where g is a monic irreducible polynomial with coefficients in the ring of integers of a p-adic field K.
Abstract: We study discrete dynamical systems of the kind h(x) = x + g(x), where g(x) is a monic irreducible polynomial with coefficients in the ring of integers of a p-adic field K. The dynamical systems of ...
3 citations