作者ntumath (math mad)
看板NTU-Exam
标题[试题] 105-2 庄武谚 代数导论二 第一次小考
时间Fri Apr 28 18:28:12 2017
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）︰106/3/9
考试时限（分钟）：30分钟
试题 :
1.(20 points) Definition: A discrete valuation ⅴon a field Ｑ is a function
ⅹ
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 ⅴon its quotient field Q such that
ⅹ
R = { x ∈ Ｑ | ⅴ(x) ≧0 )} ∪{0}.
Now let R be a DVR and Q be its quotient field. Prove that the set
ｘ
{x ∈Q | ⅴ(x) > 0 } ∪{0} is the unique maximal ideal of R.
2.(15 points) Show that the element 7∈Z[√-13] is irreducible but is
not prime.
3.(15 points) In class we have seen that the ring Z[i] of Gaussian integers
2 2
is a Euclidean domain with norm N(a+bi) = a + b . Prove that the quotient
ring Z[i]/I is finite for any nonzero ideal I of Z[i].
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