Classifying finite monomial linear groups of prime degree in characteristic zero

Let [Formula: see text] be a prime and let [Formula: see text] be the complex field. We explicitly classify the finite solvable irreducible monomial subgroups of [Formula: see text] up to conjugacy. That is, we give a complete and irredundant list of [Formula: see text]-conjugacy class representatives as generating sets of monomial matrices. Copious structural information about non-solvable finite irreducible monomial subgroups of [Formula: see text] is also proved, enabling a classification of all such groups bar one family. We explain the obstacles in that exceptional case. For [Formula: see text], we classify all finite irreducible subgroups of [Formula: see text]. Our classifications are available publicly in Magma.

Valued fields with finitely many defect extensions of prime degree

We prove that a valued field of positive characteristic [Formula: see text] that has only finitely many distinct Artin–Schreier extensions (which is a property of infinite NTP2 fields) is dense in its perfect hull. As a consequence, it is a deeply ramified field and has [Formula: see text]-divisible value group and perfect residue field. Further, we prove a partial analogue for valued fields of mixed characteristic and observe an open problem about 1-units in this setting. Finally, we fill a gap that occurred in a proof in an earlier paper in which we first introduced a classification of Artin–Schreier defect extensions.

On the compositum of orthogonal cyclic fields of the same odd prime degree

The aim of this paper is to study circular units in the compositum K of t cyclic extensions of ${\mathbb {Q}}$ ( $t\ge 2$ ) of the same odd prime degree $\ell $ . If these fields are pairwise arithmetically orthogonal and the number s of primes ramifying in $K/{\mathbb {Q}}$ is larger than $t,$ then a nontrivial root $\varepsilon $ of the top generator $\eta $ of the group of circular units of K is constructed. This explicit unit $\varepsilon $ is used to define an enlarged group of circular units of K, to show that $\ell ^{(s-t)\ell ^{t-1}}$ divides the class number of K, and to prove an annihilation statement for the ideal class group of K.