[試題] 105-2 莊武諺 代數導論二 第一次小考
課程名稱︰代數導論二
課程性質︰數學系必修
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）︰106/3/9
考試時限（分鐘）：30分鐘
試題 :
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].
--
※ 發信站: 批踢踢實業坊(ptt.cc), 來自: 140.112.25.121
※ 文章網址: https://www.ptt.cc/bbs/NTU-Exam/M.1493375294.A.4E3.html
※ 編輯: ntumath (140.112.25.105), 04/28/2017 21:31:45