New Foundation of the Theory of Regular Kummer Fields

We have seen how important a role the symbol $$ \left\{ {\frac{{v,\mu }}{l}} \right\}$$ plays in the theory of Kummer fields. The definition of this symbol in Sect. 131 and the derivation of its properties in Sect. 131 to Sect. 133 were intimately connected with the logarithmic derivative (introduced by Kummer) of the function ω(x) associated with a number ω congruent to 1 modulo ι. The computations involving the symbol $$ \left\{ {\frac{{v,\mu }}{l}} \right\}$$ in a Kummer field which were carried out in Sect. 131 to Sect. 133 correspond precisely to the considerations presented in Sect. 64 for the symbol $$ \left( {\frac{{n,m}}{2}} \right)$$ in a quadratic field. Although we have already succeeded in reducing the computational machinery invented by Kummer to modest dimensions, it still seems to me necessary, especially for the future development of the theory, to investigate whether it might not be possible to lay a foundation for the theory of Kummer fields completely lacking any computation. In this chapter I indicate briefly the way to do this.