On Normalized Integral Table Algebras (Fusion Rings) : Generated by a Faithful Non real Element of Degree 3 Introduction Splitting the Main Problem into Four Sub cases A Proof of a Non existence Sub case (2) Preliminary Classification of Sub case (2) Finishing the Proofs of the Main Results. The theory of table algebras was introduced in 1991 by Z. Arad and H.Blau in order to treat in a uniform way products of conjugacy classes and irreducible characters of finite groups. Today table algebra theory is a well established branch of modern algebra with various applications including the representation theory of finite groups algebraic combinatorics and fusion rules algebras. This book presents the latest developments in this area. Its main goal is to give a classification of the Normalized Integral Table Algebras (Fusion Rings) generated by a faithful non real element of degree 3. Divided into 4 parts the first gives an outline of the classification approach while remaining parts separately treat special cases that appear during classification. A particularly unique contribution to the field can be found in part four whereby a number of the algebras are linked to the polynomial irreducible representations of the group SL3(C). This book will be of interest to research mathematicians and PhD students working in table algebras group representation theory algebraic combinatorics and integral fusion rule algebras.