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The word "algebra" dates back about 1200 years ago to part of the title of al-Khwarizmi's book, " Kitâb al-Mukhtasar fî Hisâb al-Jabr wa'l-Muqâbala", devoted to finding solutions to practical problems which Muslims encountered in daily life concerning matters of inheritance, legacies, partition, lawsuits and commerce, with over eight hundred examples. In the 1800s, algebra became more general and more abstract as more algebraic structures like groups, quaternions, exterior algebras and vector spaces, f were invented. In this report, we present the basic concepts of finite field. Several kinds of abstract algebraic structures, particularly groups, fields, examples about infinite and finite group and examples about infinite and finite field are provided.