課程名稱︰代數導論二
課程性質︰系定必修
課程教師︰李白飛 教授
開課系所︰數學系
考試時間︰2006/04/24 10:30-12:50
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試題： Algebra II
1. Let F be an infinite field and f(x_1,...,x_n), g(x_1,...,x_n) two
polynomials in indetermines x_1,...,x_n over F with g(x_1,...,x_n)≠0.
Show that f(x_1,...,x_n)＝0. (10 pts)
4 4 4 4
2. Express x ＋y ＋z ＋w as a polynomial of the elementary symmetric
polynomials in x, y, z, w. (15 pts)
3 2
3. Solve the cubic equation x ＋6ix －6x＋2＋4i＝0 in radicals. (10 pts)
4 3 2
4. Solve the quartic equation x －6x ＋12x －6x－5＝0 in radicals. (15 pts)
5. Let f_1(x), f_2(x), f_3(x), f_4(x) and g(x) be polynomials over Ｑ such
that
5 5 2 5 3 5 2 3 4
f_1(x )＋xf_2(x )＋x f_3(x )＋x f_4(x )＝(1＋x＋x ＋x ＋x )g(x).
Show that g(x) is divisible by x－1 in Ｑ[x]. (10 pts)
6. Let n＞1 be an integer and φ_n(x) the n-th cyclotomic polynomial. Show
that φ_n(1)＝p if n is a power of a prime p and φ_n(1)＝1 otherwise.
(15 pts)
7. Find the minimal polynomial of sin6°over Ｑ. (15 pts)
3
8. Find a rational number c such that the polynomial x －9x＋c defines a
normal cubic fields. (10 pts)
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◆ From: 140.112.250.148
※ 編輯: monotones 來自: 140.112.250.148 (07/03 17:45)