Isometries of Hermitian Lattices over Discrete Valuation Rings

Abstract

In this thesis, we study the generation of the unitary group of a Hermitian lattice over a separable extension of discrete valuation rings by symmetries of the lattice. After introducing basic algebraic definitions, we give an exposition of the theory of Dedekind domains, with the goal of describing discrete valuation rings as their local rings. We then extend results about Hermitian lattices over valuation rings of non-Archimedean local fields to our setting. Finally, we extend a known result about the generation of the unitary group of certain $2$-adic Hermitian lattices to lattices over arbitrary separable extensions of discrete valuation rings, and we count the number of symmetries that appear in the factorization given by our theorem in the case where the isometry is strictly diagonalizable.