高斯整数环的一道典型习题

2023-08-20 01:20 作者:自函子 0人读过 | 我要投稿

设 $z$ 是高斯整数, 且 $%5Cgcd(%5CRe%20z%2C%5CIm%20z)%3D1$ , 证明 $%5Cmathbb%7BZ%7D%5Bi%5D%2F%5Clangle%20z%5Crangle%20$ 恰好有 $%5Cleft%7Cz%5Cright%7C%5E2$ 个元素.

证明: 不妨设其实部, 虚部为 $a%2Cb$ . 此时由 $%5Crm%7BBezout%7D$ 定理, $%5Cexists%20%5C%2C%20s%2Ct%20%5C%2C%5Cin%5C%2C%5Cmathbb%7BZ%7D$ , 满足 $as%2Bbt%3D1$ .

则有 $z(t%2Bsi)%3Dat-bs%2B%5Coverbrace%7Bas%2Bbt%7D%5E%7B1%7Di%3Dat-bs%2Bi%20%20%5Cequiv%200%20%5Cpmod%7Bz%7D%20%5Cimplies%20i%20%5Cequiv%20bs-at%20%5Cpmod%7Bz%7D%20%5C%5C$

命环同态为 $%5Cvarphi%3A%20%5Cmathbb%7BZ%7D%5Cto%5Cmathbb%7BZ%7D%5Bi%5D%2F%5Clangle%20z%5Crangle$ . 考虑 $%5Cforall%20%5C%2Cx%2Byi%5Cin%20%5Cmathbb%7BZ%7D%5Bi%5D%2C%5C%2C%5C%2Cx%2Byi%20%5Cequiv%20x%2By(bs-at)%20%5Cin%20%5Cmathbb%7BZ%7D%5Cpmod%7Bz%7D$ . 即是说像中任意元素必与一整数对应, 故 $%5Cvarphi$ 满.

$%20%5Cforall%20%5C%2C%20u%20%5Cin%5Cker%5Cvarphi%2C%5C%2C%20a%2Bbi%5Cmid%20u%20%5Cimplies%20%5Cdfrac%7Bu%7D%7Ba%2Bbi%7D%20%5Cin%20%5Cmathbb%7BZ%7D%5Bi%5D%5C%5C%0A%20%5Cimplies%20%20%20%5Cdfrac%7Bu(a-bi)%7D%7B%5Cleft%7Cz%5Cright%7C%5E2%7D%5Cin%20%5Cmathbb%7BZ%7D%5Bi%5D%20%5Cimplies%5Cleft%7Cz%5Cright%7C%5E2%20%5Cmid%20%5Cgcd(ua%2C%20ub)%20%5Cimplies%20%5Cleft%7Cz%5Cright%7C%5E2%5Cmid%20u%5Cgcd(a%2Cb)%3Du$