This chapter is about polynomials and their use. After learning the basics we discuss the concept of “secret sharing” and learn Shamir’s secret sharing scheme, which relies on the properties of polynomials over large finite fields. Then, after proving some further results on polynomials, we give a construction of a finite field whose cardinality is a power \(p^n\) of a prime p. The field constructed will be an extension of \(\mathbb Z_p\) and in this context we discuss minimal annihilating polynomials which we will need later for error-correcting coding.