Assuming the GRH and Artin conjecture for Artin L-functions, we prove that there exists a totally real number field of any fixed degree (> 1) with an arbitrarily large discriminant whose normal closure has the full symmetric group as Galois group and whose class number is essentially as large as possible. One ingredient is an unconditional construction of totally real fields with small regulato...

In this article we introduce the category of noncommutative Artin motives as well as the category of noncommutative mixed Artin motives. In the pure world, we start by proving that the classical category AM(k)Q of Artin motives (over a base field k) can be characterized as the largest category inside Chow motives which fully-embeds into noncommutative Chow motives. Making use of a refined bridg...

In 1968, M. Artin proved that any formal power series solution of a system of analytic equations may be approximated by convergent power series solutions. Motivated by this result and a similar result of Płoski, he conjectured that this remains true when we replace the ring of convergent power series by a more general ring. This paper presents the state of the art on this problem, aimed at non-...

We say that a prime number $p$ is an $\textit{Artin prime}$ for $g$ if mod generates the group $(\mathbb{Z}/p\mathbb{Z})^{\times}$. For appropriately chosen integers $d$ and $g$, we present conjecture asymptotic $\pi_{d,g}(x)$ of primes $p \leq x$ such both $p+d$ are Artin $g$. In particular, identify class pairs $(d,g)$ which $\pi_{d,g}(x) =0$. Our results suggest distribution pairs, amongst o...

In this note, we apply the group-theoretic method to study Artin’s conjecture, and introduce the notations of nearly nilpotent groups and nearly supersolvable groups to answer of a question of Arthur and Clozel. As an application, we show that Artin’s conjecture is valid for all nearly supersolvable Galois extensions of number fields as well as all solvable Frobenius extensions.