Polynomials arise naturally in many areas of mathematics, physics and engineering, which is a basic research topic in algebra (Fuhrmann in A Polynomial Approach to Linear Algebra. Springer, New York, 1996 [1]; Weiße et al. in Reviews of Modern Physics 78:275–306, 2006 [2]). This chapter introduces the fundamental theory of Boolean polynomial over the Boolean field \(\mathcal {B}=\{0,1\}\) . Firstly, the concept of unary Boolean polynomial is proposed. After that, the division and greatest common divisor of Boolean polynomial are well discussed. Finally, the factorization theorem of Boolean polynomial is revealed.