Chapter 1

Abstract Algebra :

Groups, Ring s an d

Fields

This course will aim at understandin g convex poly topes, which

damental geometri c object s i n combinatorics , usin g techniqu

algebra an d discret e geometry . Polytope s aris e everywher

real worl d an d i n mathematics . Th e mos t famou s example s

Platonic solid s i n three-dimensiona l space : cube, tetrahedron,

dron, icosahedron and dodecahedron, which were known to the

Greeks. Th e natural first approac h to understanding polytope

be through geometr y a s they ar e first an d foremos t geometri c

However, an y experienc e wit h visualizin g geometri c object s

you soo n tha t geometr y i s alread y quit e har d i n three-dim

space, an d i f on e ha s t o stud y object s i n four - o r higher-dim

space, the n i t i s essentiall y hopeles s t o rel y onl y o n ou r ge

and drawin g skills . Thi s frustratio n le d mathematician s t o

covery tha t algebr a ca n b e use d t o encod e geometr y and , sin

bra doe s no t suffe r fro m th e sam e limitation s a s geometr y

ing wit h highe r dimensions , i t ca n serv e ver y wel l a s th e l

of geometry . A simpl e exampl e o f thi s translatio n ca n b e

noting that , whil e i t i s har d t o visualiz e vector s i n four-dim