This chapter is about polynomials and their use. After learning the basics we discuss Lagrange’s interpolation needed for Shamir’s secret sharing scheme that we discuss in Chap. 6. Then, after proving some further results on polynomials, we give a construction of a finite field whose cardinality \(p^n\) which is a power of a prime p. This field is constructed as polynomials modulo an irreducible polynomial of degree n. The field constructed will be an extension of \(\mathbb Z_p\) and in this context we discuss minimal annihilating polynomials which we will need in Chap. 7 for the construction of good error-correcting coding.