The determination of the algebraic representations of coordinates of finite field elements with the elements themselves as the variables

which is a new property of finite fields

is investigated.Based on the partition of equivalent classes

the resolving of a linear system of equations and the calculation of the dual basis of the standard basis

three methodologies are presented.With those results

we have successfully given an essential explanation to the simplicity of the algebraic representation of Rijndael S-box and provided a direct proof to the equivalence between any two coordinate functions of Rijndael S-box.