합성 Hurwitz 다항식환 h(Z,Q)의 고차 기약 다항식 분류

Abstract

Let R be a commutative ring with identity, X be an indeterminate over R, and R[X] be the ring of polynomials over R. Keigher introduce a variation of usual ring of polynomials over R, called the ring of Hurwitz polynomials over R. The ring of Hurwitz polynomials over R is denoted by h(R). For an extension R ⊆ D of commutative rings with identity, the ring R+XD[X], called the ring of composite polynomials, is between R[X] and D[X]. By a variation of usual composite polynomial, we consider the ring of composite Hurwitz polynomials, denoted by h(R,D), between two rings h(R) and h(D) of Hurwitz polynomials over R and D, respectively. Let Z and Q be the ring of integers and the field of rational numbers, respectively. In this thesis, we study when composite Hurwitz polynomials of h(Z,Q) with high degrees are irreducible. More precisely, we give a necessary and sufficient condition for composite Hurwitz polynomials of degree 4 and 5 in h(Z,Q) to be irreducible by using a relation between composite Hurwitz polynomials in h(Z,Q) and usual polynomials in Q[X].