### A Bound for the Least Prime Ideal in the Chebotarev Density Theorem.

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This article is a short version of the paper published in J. Number Theory 145 (2014) but we add new results and a brief discussion about the Torsion Conjecture. Consider the family of superelliptic curves (over ℚ) ${C}_{q,p,a}:{y}^{q}={x}^{p}+a$, and its Jacobians ${J}_{q,p,a}$, where 2 < q < p are primes. We give the full (resp. partial) characterization of the torsion part of ${J}_{3,5,a}\left(\mathbb{Q}\right)$ (resp. ${J}_{q,p,a}\left(\mathbb{Q}\right)$). The main tools are computations of the zeta function of ${C}_{3,5,a}$ (resp. ${C}_{q,p,a}$) over ${}_{l}$ for primes l ≡ 1,2,4,8,11 (mod 15) (resp. for primes l ≡ -1 (mod qp))...

Let Γ ⊂ ℚ * be a finitely generated subgroup and let p be a prime such that the reduction group Γₚ is a well defined subgroup of the multiplicative group ₚ*. We prove an asymptotic formula for the average of the number of primes p ≤ x for which [ₚ*:Γₚ] = m. The average is taken over all finitely generated subgroups $\Gamma =\u27e8a\u2081,...,{a}_{r}\u27e9\subset \mathbb{Q}*$, with ${a}_{i}\in \mathbb{Z}$ and ${a}_{i}\le {T}_{i}$, with a range of uniformity ${T}_{i}>exp\left(4{\left(logxloglogx\right)}^{1/2}\right)$ for every i = 1,...,r. We also prove an asymptotic formula for the mean square of the error terms in the asymptotic formula with a similar...

We extend the "character sum method" for the computation of densities in Artin primitive root problems given by Lenstra and the authors to the situation of radical extensions of arbitrary rank. Our algebraic set-up identifies the key parameters of the situation at hand, and obviates the lengthy analytic multiplicative number theory arguments that used to go into the computation of actual densities. It yields a conceptual interpretation of the formulas obtained, and enables us to extend their range...

In this paper, we give asymptotic formulas for the number of cyclic quartic extensions of a number field.

For each transitive permutation group $G$ on $n$ letters with $n\le 4$, we give without proof results, conjectures, and numerical computations on discriminants of number fields $L$ of degree $n$ over $\mathbb{Q}$ such that the Galois group of the Galois closure of $L$ is isomorphic to $G$.