Date of Award

8-1-2017

Degree Name

Doctor of Philosophy

Department

Mathematics

First Advisor

Ban, Dubravka

Abstract

--------------------------------------------------------------- % The "*" following chapter or section commands omits chapter/ % section numbers. It also does not include the chapter/section % in the table of contents -- the \addcontentsline can be used % to manually force its entry. %--------------------------------------------------------------- %%%%%% %%%%%% This page needed only if Thesis or Dissertation %%%%%% Change title of this page to reflect type of paper %%%%%% %% Change title below to reflect Thesis or Dissertation \chapter*{AN ABSTRACT OF THE DISSERTATION OF } \addcontentsline{toc}{chapter}{Abstract} % DATE below is the date of your defense % Change those words in all caps below Abdulkarem Alhuraiji, for the Doctor of Philosophy degree in MATHEMATICS, presented on May 11, 2017, at Southern Illinois University Carbondale. \vspace{14pt} \noindent TITLE: COUPLING OF QUADRATIC LATTICES % Change title above and put all in CAPS \vspace{14pt} \noindent MAJOR PROFESSOR: Dr. A. Earnest \vspace{14pt}\\ The subject of this dissertation is the representation theory of quadratic lattices over the ring of rational integers and its $p$-adic completions. The main result that will be obtained is the following. If any two integral lattices of rank $n$, where $n\geq 3$, represent the same lattices of rank $(n-1)$ (i.e., those of codimension 1) then they are isometric. In order to facilitate the study of such representation problems, in this paper a notion of coupling of quadratic lattices is introduced and studied. \chapter*{ACKNOWLEDGMENTS} \addcontentsline{toc}{chapter}{Acknowledgments} I would like to thank and appreciate to Dr. Andrew Earnest for his invaluable assistance, wisdom, patience, guidance, generosity, support, enthusiasm and encouragement for writing this dissertation. I really cannot explain my feeling how Dr. Earnest helpful, supportable and understandable what I can say ``Thank you from my heart''. Also my sincere thanks goes to Dr. Dubravka Ban, Dr. Henri Schurz, Dr. Jerzy Kocik and Dr. Mohammad Sayeh for their guidance and their patience in serving on my committee. Thanks are extended to all the wonderful members in the Department of Mathematics. A special thanks and appreciation to all my family.

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