합성 Hurwitz 다항식환 h(ℤ, ℚ)의 기약성 연구

Abstract

Let 𝑅⊆𝐷 be an extension of commutative rings with identity, and 𝑥 be an indeterminate over 𝐷. Let 𝑅[𝑥](respectively, 𝐷) be the ring of polynomials over 𝑅(respectively, 𝐷). The ring 𝑅＋𝐷[𝑥], called the composite polynomial ring, is between 𝑅[𝑥] and 𝐷[𝑥]. As a generalization of usual polynomial over 𝑅, Hurwitz polynomial over 𝑅 was introduced by Keigher. Let h(𝑅) and h(𝐷) be the rings of Hurwitz polynomials over 𝑅 and 𝐷, respectively. We introduce the composite Hurwitz polynomial ring h(𝑅, 𝐷) between h(𝑅) and h(𝐷). In this thesis, we study when an element of h(ℤ, ℚ) is irreducible, where ℤ and ℚ are the ring of integers and the field of rational numbers, respectively. By using a relation between composite Hurwitz polynomials in h(ℤ, ℚ) and usual polynomials in ℚ[𝑥], we give a necessary and sufficient condition for composite Hurwitz polynomials in h(ℤ, ℚ) to be irreducible.