A numerical semigroup S is the subset of non-negative integers that is often expressed in terms of its generators, S = ⟨g1, . . . , gk⟩. A numerical semigroup of the form Sm = ⟨m, m + 1, . . . , 2m − 1⟩ has maximal embedding dimension and a numerical semigroup algebra of Sm over a field F is the set of polynomials consisting only of terms xi with i ∈ S. In this thesis, we explore the reducibility of polynomials over numerical semigroup algebras. We will present improved upper bounds for the longest factorization length in Fq[x] of an irreducible element in Fq[Sm].
