課程名稱︰代數導論二
課程性質︰必修
課程教師︰莊武諺
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2014/03/27
考試時限（分鐘）：30分鐘
試題 :
代數導論第三次小考
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.