设 F 是任意域,M_n 记 F 上 n×n(n≥2)矩阵全体构成的乘法半群.熟知,行列式映射是 M_n 到 F 的乘法同态.本文考虑其反问题,即决定全部从 M_n 到 F 的半群乘法同态,亦即 M_n 的全部积性函数.我们以 Hom(M_n,F)记 M_n 到 F 的乘法同...
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设 F 是任意域,M_n 记 F 上 n×n(n≥2)矩阵全体构成的乘法半群.熟知,行列式映射是 M_n 到 F 的乘法同态.本文考虑其反问题,即决定全部从 M_n 到 F 的半群乘法同态,亦即 M_n 的全部积性函数.我们以 Hom(M_n,F)记 M_n 到 F 的乘法同态全体构成的集,即若(?)∈Hom(M_n,F),则有(?)(AB)=(?)(A)(?)(B) (?)A、B∈M_n又我们用 GL_n(F)及 SL_n(F)记 F 上一般线性群与特殊线性群.I_n 记 M_n 中单位阵,E_(ij)记 M_n 中(i,j)位置是1,其余位置是0的矩阵。当λ为 F 中非零元素时,F_(ij)(λ)
For equation(1) af(ri)=bg(n), where a, b are given positive integers! f(n), g(n) are multiplicative functions, and f(n).g(n), we prove that1. If f(n), g(n) satisfy conditions(i) For given prime’p and N, there is a po...
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For equation(1) af(ri)=bg(n), where a, b are given positive integers! f(n), g(n) are multiplicative functions, and f(n).g(n), we prove that1. If f(n), g(n) satisfy conditions(i) For given prime’p and N, there is a positive integer UN, which only depent on N, when a-.aN,i/(p') j/or\g(p.1\\(ti) For given positive integer or, there is a positive integer (3a> when o? l.r.r- l(or l.<7Oa) l\\ >c> 1, where c is a constand, then (1) has only /finitely many positive integer solutions. 3. The equationsand ahave only finitely many solutions.3. We give also, in the present note, all the positive integral solutions of the following equations:=dz(n),and
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