This paper is concerned with the construction of sum and product tables for Galois fieldsGF(q)in whichqis a power of a prime numberp.Sufficient elementary theory is presented to provide a basis for the development of methods of representation, addition and multiplication for the field elements of GF(q).The methods are oriented towards the use of operations from the prime field GF(p)that are easy to define and implement in terms of modulo‐Parithmetic. They lead to a compact form of representation for the field elements and to simply applied procedures for the construction of the sum and product tables. Examples of the methods and procedures proposed are given throughout, and sum and product tables are given for Galois fieldsGF(q)whenqhas values up to 27.