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）︰2007.04.26
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 中 考 試 題 2007/4/26
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 α＝ √2^(1/3)－1, β＝ √30＋10‧5^(1/2). Find two monic irreducible
polynomials f(x), g(x) 屬於 Ｑ[x] such that f(α)＝0, g(β)＝0. (20％)
2 3 _
二、Prove that x －5 is irreducible over Ｑ[√2]. (15％)
三、Let K be a finite extension field of a field F and D be a subring of K
such that F 包含於 D 包含於 K. Prove or disprove that D is a field. (15％)
__
四、Let i＝√-1, u＝2＋5i and v＝3＋i. Find four intergers a, b, c, d 屬於 Ｚ
2 2
such that u＝v(a＋bi)＋(c＋di) and c ＋d ＜10. (20％)
3 3
五、Let f(x)＝x －1, g(x)＝x －2 be two polynomials in Ｚ_p[x], where p is an
odd prime and Ｚ_p 包含於 F, Ｚ_p 包含於 K are field extensions such that
there are α_1, α_2, α_3 屬於 F, β_1, β_2, β_3 屬於 K
3
f(x)＝x －1＝(x－α_1)(x－α_2)(x－α_3), F＝Ｚ_p[α_1,α_2,α_3],
3
g(x)＝x －2＝(x－β_1)(x－β_2)(x－β_3), K＝Ｚ_p[β_1,β_2,β_3].
(1) Suppose that p≠1 (mod 3). What are F and K ? Prove your answer.
(2) Suppose that p＝7. What are F and K ? Prove your answer.
(3) Suppose that p＝31. What are F and K ? Prove your answer. (30％)
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 140.112.7.59
※ 編輯: monotones 來自: 140.112.7.59 (07/02 21:29)