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Uniformity over primes of unramified splittings. (English) Zbl 0908.11054

Given a fixed number field \(K\), the Chebotarev density theorem asserts that the set of rational primes \(p\) having a prescribed splitting behaviour in the ring of integers of \(K\) has a natural density that can be expressed as a relative frequency of certain elements in the Galois group of the natural closure of \(K\). Thus, in the Chebotarev density theorem, \(K\) is fixed and \(p\) varies.
In the converse Chebotarev problem we ask for the ‘density’ of number fields in which a fixed prime \(p\) has a prescribed splitting behaviour. It seems natural to consider all fields of a fixed degree \(n\), and order them according to the size of the discriminant. With the appropriate definition of density, it was shown by the authors in an earlier paper [I. Del Corso and R. Dvornicich, J. Number Theory 45, 28-44 (1993; Zbl 0779.11047)] that such a converse density exists for all possible splittings and that, by taking the limit for \(p\to\infty\), one finds the density predicted by Chebotarev’s density theorem. Moreover, when considering the case \(n= 3\) in detail, the authors found that each density is uniformly described by the value at \(p\) of a rational function \(r_n(x)\) of \(\mathbb{Q}(X)\) and conjectured that this is true for any \(n\).
In this paper, the authors prove their conjecture for all \(n\) and all unramified splittings. Moreover, they show that \(r_n(x)\) is effectively computable by recursive formulas contained in the proofs, and has a denominator that is a product of cyclotomic polynomials only. The proof proceeds by expressing the required density in terms of certain integrals over suitable \(p\)-adic domains with the Haar measure of \(\mathbb{Z}^*_p\). In the case of unramified splitings, these integrals become integrals on all \(\mathbb{Z}^n_p\) and have a special form that allows them to be computed by an inductive argument.

MSC:

11R45 Density theorems

Citations:

Zbl 0779.11047
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References:

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