Reducibility of symmetric polynomials

This thesis investigates the reducibility of trivariate homogeneous symmetric polynomials. For polynomials of degrees 2, 3, 4 and 5 we have complete results. Specifically, we classify all the possible factorizations of such polynomials and give conditions on the coefficients of these polynomials that determine which factorizations occur. The polynomials of this thesis have indeterminates x, y, and z. The symmetric group on these three indeterminates acts on these polynomials by permuting the indeterminates. Symmetric polynomials are left unchanged by this group action. If a polynomial is reducible, then the group acts on the factorization, permuting the factors. The unique factorization property of polynomials allows us to establish which factorizations are possible. Once all possible factorizations are known, we then specify the conditions on the coefficients of the polynomials under which each of these possible factorizations occurs