课程名称︰代数导论二
课程性质︰数学系必修
课程教师︰庄武谚
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰106/3/23
考试时限（分钟）：30分钟
试题 :
You may assume Gauss's Lemma.
1.(25 points) Let R be an integral domain with quotient field F and let
p(x) be a monic polynomial in R[x]. Assume that p(x) = a(x)b(x)
where a(x) and b(x) are nonconstant monic polynomials in F[x] of
smaller degrees. Prove that if ﹁(a(x)∈R[x]) then R is not a UFD.
2.(25 points) Prove that if f(x) and g(x) are polynomials with rational
coefficients and p(x) = f(x)g(x) has integer coefficients, then the
product of any coefficient of f(x) with any coefficients of g(x) is
an integer.