课程名称︰代数导论二
课程性质︰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）︰2014/03/27
考试时限（分钟）：30分钟
试题 :
代数导论第三次小考
INTRODUCTION TO ALGEBRA II - QUIZ III
MAR 27 2014
(1) (50 points) First we recall Gauss' Lemma. Let R be an Unique Factorization
Domain (UFD) with quotient field F and let p(x) ∈ R[x]. If p(x) = A(x)B(x)
for some nonconstant polynomials A(x),B(x) ∈ F[x], then there exist r,s∈F
such that rA(x) = a(x) and sB(x) = b(x) both are in R[x] and p(x)=a(x)b(x)
is a factorization in R[x].
Assuming Gauss' Lemma, please show that R is an UFD if and only if R[x]
is an UFD.