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）︰2014/03/13
考試時限（分鐘）：30分鐘
試題 :
代數導論第二次小考
INTRODUCTION TO ALGEBRA II - QUIZ II
Mar 13 2014
(1) (15 points) Prove that a Euclidean domain is a principal ideal domain(PID).
(2) (15 points) Prove that if R is a PID and D is a multiplicatively closed
subset of R, then (D^-1)R is also a PID.
(3) (15 points) Definition: A discrete valuation v on a field Q is a function
×
v:Q → Z satisfying: (1) v(xy) = v(x) + v(y), (2) v is surjective, and
(3) v(x+y)≧min{v(x),v(y)}. Definition: An integral domain R is called a
discrete valuation ring (DVR) if there exists a discrete valuation v on its
quotient field Q such that R = {x∈Q|v(x)≧0} ∪ {0}.
Now Let Γ be a DVR and Φ be its quotient field. Prove that the set
{x∈Φ|v(x)>0} ∪ {0} is the unique maximal ideal of Γ.
(4) (15 points) Prove that a DVR is a Euclidean domain.