Abstract:
We fix a prime p and let f(X) vary over all monic integer polynomials of fixed degree n. Given any possible shape of a tamely ramified splitting of p in an extension of degree n, we prove that there exists a rational function φ(X)∈ℚ(X) such that the density of the monic integer polynomials f(X) for which the splitting of p has the given shape in ℚ[X]/f(X) is φ(p) (here reducible polynomials can be neglected). As a corollary, we prove that, for p≥n, the density of irreducible monic polynomials of degree n in ℤ p [X] is the value at p of a rational function φ n (X)∈ℚ(X). All rational functions involved are effectively computable.
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Received: 15 September 1998 / Revised version: 21 October 1999
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Del Corso, I., Dvornicich, R. Uniformity over primes of tamely ramified splittings. manuscripta math. 101, 239–266 (2000). https://doi.org/10.1007/PL00005848
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DOI: https://doi.org/10.1007/PL00005848