Antonella Perucca

Let be a number field and . Artin's primitive root conjecture concerns the proportion of primes of for which is well defined, non‐zero, and generates the multiplicative group at . The conjecture states that the set of those primes admits a density and provides a formula for it. Conditionally under the generalized Riemann hypothesis, the conjecture was proven over (for the integers) by Hooley in 1967 and over any number field by Cooke and Weinberger in 1975. Hooley also provided a closed formula over , describing the rational number that is the ratio between the density and the Artin constant. In this paper, we provide such a closed formula over any number field.

Let K be a number field, and let G be a finitely generated subgroup of K × . For every prime number ℓ and for all positive integers m , n with m ⩾ n we show that the structure of the Galois group of the Kummer extension K ( ζ ℓ m , G ℓ n ) / K ( ζ ℓ m ) only depends on G through parameters that express divisibility properties over K (respectively, over K ( ζ 4 ) if ℓ = 2 , ζ 4 ∉ K , m ≥ 2 ). Moreover, we describe an explicit finite procedure to compute at once the Galois group structure for all extensions K ( ζ M , G N ) / K ( ζ M ) with M , N positive integers such that N ∣ M . Our work builds on results in Kummer theory by the last-named author joint with Debry, Hörmann, Perissinotto, Sgobba, and Tronto.

We consider Artin's conjecture on primitive roots over a number field , reducing an algebraic number . Under the Generalised Riemann Hypothesis, there is a density counting the proportion of the primes of for which is a primitive root. This density is a rational multiple of an Artin constant that depends on the largest integer such that . The aim of this paper is bounding the ratio , under the assumption that . Over , this ratio is between and 2, these bounds being optimal. For a general number field , we provide upper and lower bounds that only depend on .

We explore the geometry of the renowned Vichten Roman Mosaic. The central section of the mosaic features a tessellation composed of octagons, half-squares, and symmetrical pentagons. We present a Sangaku-style exercise to investigate these polygons. Additionally, we examine the whirl patterns of the mosaic and the optical illusion they create. Our activities offer a mathematical exploration that requires only basic school-level mathematics, highlighting the close connection between mathematics and art.