Campana points and powerful values of norm forms

We give an asymptotic formula for the number of weak Campana points of bounded height on a family of orbifolds associated to norm forms for Galois extensions of number fields. From this formula we derive an asymptotic for the number of elements with m-full norm over a given Galois extension of $$\mathbb {Q}$$ Q . We also provide an asymptotic for Campana points on these orbifolds which illustrates the vast difference between the two notions, and we compare this to the Manin-type conjecture of Pieropan, Smeets, Tanimoto and Várilly-Alvarado.

Galois covers of ℙ1 and number fields with large class groups

For each finite subgroup [Formula: see text] of [Formula: see text], and for each integer [Formula: see text] coprime to [Formula: see text], we construct explicitly infinitely many Galois extensions of [Formula: see text] with group [Formula: see text] and whose ideal class group has [Formula: see text]-rank at least [Formula: see text]. This gives new [Formula: see text]-rank records for class groups of number fields.

Constructing totally p-adic numbers of small height

Bombieri and Zannier gave an effective construction of algebraic numbers of small height inside the maximal Galois extension of the rationals which is totally split at a given finite set of prime numbers. They proved, in particular, an explicit upper bound for the lim inf of the height of elements in such fields. We generalize their result in an effective way to maximal Galois extensions of number fields with given local behavior at finitely many places.

Galois extensions for quasigroupoid magmas

In this paper, we extend the result proved by Ulbrich about the characterization of Galois extensions linked to group algebras upon the non-associative (quasigroup) quasigroupoid magma setting. Also, as a particular instance of the results contained in this paper, we obtain the ones proved for Galois extensions related with groupoid algebras.