We have obtained Artin-Hasse formula if we take the prime π=λ for the norm residue symbol (υ,μ), making use of the properties of the norm residue symbol and congruence classes, we prove an important congruence in a direct product decomposition of quotient group for the element κ_a of representative, by the calculation of making the conclusions we obtain the Takagi formula and Takagi-Hasse formula.
对于范数剩余符号(υ,μ),若取素元π=λ已经得到了Artin Hasse公式,而在此利用范数剩余符号的性质及剩余类合同式的性质,证明商群的直积分解中的代表元κa的一个重要合同式,并利用此结论通过计算得到高木公式,由生成函数公式进而得到Takagi Hasse公式。