代数学讲稿3-6.ppt

Chapter III: Rings Contents 1. Rings and Homomorphisms 2. Ideals 3. Factorization in Commutative Rings 4. Rings of Quotients and Localization 5. Rings of Polynomials and Formal Power Series 6. Factorization in Polynomial Rings §3.6 Factorization in  Polynomial Rings Main Points Consider the topics introduced in Section 3 in the context of polynomial rings over a commutative ring. Begin with two basic tools: the concept of the degree of a polynomial and the division algorithm. Main Points (Cont.) Factors of degree one of a polynomial are then studied; finding such factors is equivalent to finding roots of the polynomial. Consider irreducible factors of higher degree: Eisenstein’s irreducibility criterion is proved and it is shown that the polynomial domain D[x1,…,xn] is a unique factorization domain if D is. Degree of A Monomial Let R be a ring. The Degree of a nonzero monomial ax1k1…xnkn∈R[x1,…,xn] is the nonnegative integer k1+…+kn. If f is a nonzero polynomial in R[x1,…,xn], then by Thm 5.4. Degree of A Polynomial The (total) degree of the polynomials f is the maximum of the degrees of the monomials s.t. ai≠0 (i=1,…,m). The (total) degree of f is denoted deg f. Remarks A nonzero polynomial f has degree zero iff f is a constant polynomial. A polynomial which is a sum of monomials, each of which has degree k, is said to be homogeneous of degree k. Review For each k (1≤k≤n), R[x1,…,xk-1,xk+1,…, xn] is a subring of R[x1,…,xn]. The degree of f in xk is the degree of f considered as a polynomial in one indeterminate xk over the ring R[x1,…,xk-1, xk+1,…,xn]. Example The polynomial 3x12x22x32+3x1x34-6x23x3∈Z[x] has degree 2 in x1, degree 3 in x2, degree 4 in x3 and total degree 6. Deg 0 Define the degree of the zero polynomial to be -∞ and Adopt the following conventions about the symbol deg 0=-∞: -∞n, (-∞)+n=-∞=n+(-∞) for every integer n; (-∞)+(-∞)=-∞. Arithmetic of Degrees Theorem 6.1 Let R be a ring and f,