The Structure Of Quasi-reducible Polynomials Over Prime Field

This paper is mainly focus on the structure and properties of irreducible polynomial over the prime field P. Based on the well known results of irreducible polynomial, we introduce the concept of surjection. After studying on the surjective polynomial and irreducible polynomial, we discover a new class of polynomial named quasi-reducible polynomial.Further, we also show some more results such as Fermat theorem on polynomial and properties of cyclic polynomial.Suppose n is a positive integer,where a₁,a₂,…,a_n are different integers,φ(x) is polynomial of integral coefficient. If deg(φ(x)) = r≤n , then f(x) is called quasi-reducible polynomial.By studying of quasi-reducible polynomial we get elementary results on polynomials over prime fields, for example:(1) When n≥5 and deg(φ(x))1,a₂,…,a_n are different integers,φ(x) is polynomial of integral coefficient.If deg(φ(x)) < [(?)],then f(x) is called strongquasi-reducible polynomial.In this thesis we also get several new characterization on strong quasi-reducible polynomial. For example, ifandφ(x) =1 is a strong quasi-reducible polynomial,then f(x) is reducible over field Q iff a₁,a₂,a₃,a₄ are four consecutive integers.Further more, we propose the concept of strong quasi-reducible polynomial of order k.A polynomial is called strong quasi-reducible polynomial of order k ifwhere a₁,a₂,…,a_n are different integers, n and k are positive integers,φ(x) is polynomial of integral coefficient, and deg(φ(x)) < [(?)].For k=2, i.e., strong quasi-reducible polynomial of order 2, we also get some conclusion which as follow:(1) Supposeis a trong quasi-reducible polynomial of order 2, if exist at least (?) positive integers a_k in {a₁,a₂,…,a_n},such that |a_k|>1,and every distance between different a_j,a_k are greater than 2, then f(x) is irreducible over field Q.(2) Supposeis a quasi-reducible polynomial of order 2, where a_i are nonzero integers,i =1,2,…,n.If 12≤n and deg(f(x))<48,then f(x) is irreducible over field Q.At last, we investigate polynomials over Euclidean domains and study at quasi-reducible polynomials. Many known theorems could be extended to Euclidean domain,such as Fermat theorem on polynomials. In other word, we prove that:If f(x),g(x),h(x) are polynomials over an Euclidean domain D, and (f(x),g(x),h(x))-1 such that not all of them are constant, then f(x)ⁿ + g(x)ⁿ=h(x)ⁿ will never be true for all integers n≥3.