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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 probability that a random monic polynomial of degree n with coefficients in the ring Z p of p-adic integers splits over Z p into linear factors was calculated.
Abstract: Let n be a positive integer and let p be a prime. We calculate the probability that a random monic polynomial of degree n with coefficients in the ring Z_p of p-adic integers splits over Z_p into linear factors.
4 citations
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TL;DR: In this article, the class group of a number field measures the failure of unique factorization in its ring of integers, and the structure of all irreducible factorizations of an element in the ring of the integers of the number field is given.
Abstract: We give a precise description of how the class group of a number field measures the failure of unique factorization in its ring of integers. Specifically, following ideas of Kummer, we determine the structure of all irreducible factorizations of an element in the ring of integers of a number field, and give a combinatorial description for the number of such factorizations. In certain cases, we show how quadratic forms can explicitly provide all such factorizations.
4 citations
TL;DR: In this paper, the authors estimate the number of points of bounded height in a Hermitian vector bundle over a number field and its ring of integers, with respect to the anticanonical line bundle.
Abstract: Let K be a number field and \(\) its ring of integers. Let \(\) be a Hermitian vector bundle over \(\). In the first part of this paper we estimate the number of points of bounded height in \(\) (generalizing a result by Schanuel). We give then some applications: we estimate the number of hyperplanes and hypersurfaces of degree d>1 in \(\) of bounded height and containing a fixed linear subvariety and we estimate the number of points of height, with respect to the anticanonical line bundle, less then T (when T goes to infinity) of ℙNK blown up at a linear subspace of codimension two.
4 citations
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TL;DR: In this article, a random walk on the ring of integers mod n is analyzed, and the existence of a total variation cutoff for this process is proved, with cutoff time dependent on whether the step distribution has zero mean.
Abstract: We analyse a random walk on the ring of integers mod $n$, which at each time point can make an additive 'step' or a multiplicative 'jump'. When the probability of making a jump tends to zero as an appropriate power of $n$ we prove the existence of a total variation cutoff for this process, with cutoff time dependent on whether the step distribution has zero mean.
4 citations
TL;DR: In this article, a variant of Karoubi's relative Chern character for smooth, separated schemes over the ring of integers in a p-adic field was constructed and compared with the rigid syntomic regulator.
Abstract: We construct a variant of Karoubi's relative Chern character for smooth, separated schemes over the ring of integers in a p-adic field and prove a comparison with the rigid syntomic regulator. For smooth projective schemes we further relate the relative Chern character to the etale p-adic regulator via the Bloch-Kato exponential map. This reproves a result of Huber and Kings for the spectrum of the ring of integers and generalizes it to all smooth projective schemes as above.
4 citations