作者ntnusliver (炸虾大叔~~)
看板Math
标题Re: [代数] 有关 Cyclotomic polynomial 的问题
时间Wed Sep 9 20:12:01 2020
※ 引述《TimcApple (肥鹅)》之铭言：
: (Def)
: Let zeta_n = exp(i 2pi/n), n-root of unity
: Let Phi_n(x) be the minimum poly of zeta_n over Q
: (Problem) Prove or disprove:
: Let g(x) in Z[x], deg(g) = n-1, with all coefficient nonnegative, n > 1.
: If g(zeta_n) = 0, then g(x) have periodic coefficient,
: which means g(x) = h(x) (1 + x^T + ... x^(kT)) for some k, T in N.
: Ex: Write (a_0, a_1, a_2, ...) instead of a_0 + a_1 x + a_2 x^2 + ...
: n = 6, g(x) = (1,2,3,1,2,3), g(zeta_6) = 0
给你看一个例子
考虑 n=6 , zeta_6 , phi_6(x)=x^2 -x +1
g(x)=x^5 + x^4 +x^3 + 2x^2 + 0x +2 符合deg(g)=5
=(x^2 -x +1)(x^3 +2x^2 +2x +2) (这两个因式皆irreducible)
g(zeta_6)=0 但g(x)的因式均不为(1 + x^T + ... x^(kT)) 的形式
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1F：推 TimcApple : 感谢ow o 09/09 22:09
2F：→ TimcApple : 那大概没有明显的结果了qw q 09/09 22:09