Unitary groups and ramified extensions

TL;DR

This paper classifies skew-hermitian forms over local rings and investigates properties of associated unitary groups, including their orders in finite cases, expanding understanding of these algebraic structures.

Contribution

It provides a comprehensive classification of skew-hermitian forms over certain local rings and analyzes fundamental properties of the related unitary groups, including order calculations.

Findings

Classified all non-degenerate skew-hermitian forms over specific local rings.

Determined the orders of unitary groups over finite rings.

Explored fundamental properties of these unitary groups.

Abstract

We classify all non-degenerate skew-hermitian forms defined over certain local rings, not necessarily commutative, and study some of the fundamental properties of the associated unitary groups, including their orders when the ring in question is finite.