Intersective polynomials, that is, polynomials in Z[x] with an integer root in any modulus, have many applications in Galois Theory. Some examples of intersective polynomials include any polynomial in Z[x] with an integer root. Some more complicated examples exist, such as the polynomial (x_ -- 19)(x_ + x + 1). In this paper, we explore the inner workings of intersective polynomials, using only elementary number theory. We show how to create intersective polynomials by multiplying non-intersective polynomials with certain properties. We also discuss intersective polynomials as a semigroup and factorization properties
