課程名稱︰代數導論二
課程性質︰系定必修
課程教師︰黃漢水 老師
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2007.06.21
考試時限（分鐘）：180 分
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題： 代 數 導 論 (二) 期 末 考 試 題 2007/6/21
Notation: Ｚ is the set of all intergers,
Ｑ is the set of all rational numbers,
Ｒ is the set of all real numbers and
Ｃ is the set of all complex numbers.
一、Let s_1＝y_1＋y_2＋y_3, s_2＝y_1‧y_2＋y_2‧y_3＋y_1‧y_3,
s_3＝y_1‧y_2‧y_3 be the elementary symmetric functions in
Ｑ[y_1, y_2, y_3]. Express
3 3 3
y_1(y_2＋y_3)＋y_2(y_1＋y_3)＋y_3(y_1＋y_2) as a rational function of
s_1, s_2, s_3. (20％)
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二、Let α＝√(1/2)^(1/2)＋(-1/2)^(1/2). (20％)
(1) Find a monic irreducible polynomials f(x) 屬於 Ｑ[x] such that f(α)＝0.
(2) Let Ｋ be the splitting field of f(x) over Ｑ. Find [K:Ｑ].
(3) Descibe the group Ｇ(Ｋ∕Ｑ).
三、Prove or disprove that there are fields F, K such that K is finite normal
extension of F and Ｇ(K∕F) is a cyclic group of order 7. (20％)
2
四、Let F＝Ｚ_3, f(x)＝x －x－1 屬於 Ｚ_3[x], E＝Ｚ_3[x]∕(f(x)) be a finite
4
field with |E|＝9 and g(x)＝x －2 屬於 E[x]. If K is the splitting field
of g(x) over E.
(1) Describe the group Ｇ(K∕E).
(2) Describe the group Ｇ(K∕F). (20％)
4 2
五、Let f(x)＝x ＋(3－6i)x ＋(－8－6i) 屬於 Ｃ[x].
Solve the equation f(x)＝0. (20％)
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◆ From: 140.112.7.59
※ 編輯: monotones 來自: 140.112.7.59 (07/02 21:30)