It is well known that the algebraic expression of the Advanced Encryption Standard (AES) Rijndael S-box is very simple and only 9 terms are involved. In this paper a new algebraic method is proposed to illustrate the reason why AES S-box algebraic expression is very simple, which corrects all the available inaccurate illustrations. An improved AES S-box is also presented to improve the complexity of AES S-box algebraic expression with terms increasing from 9 to 255 and algebraic degree invariable. The improved AES S-box also has good properties of Boolean functions in SAC and balance, and is capable of attacking against differential cryptanalysis with high reliable security.