# Artin L- Function properties

Hi, I'm trying to understand the proof of one of the properties of the Artin L-function. I have the following doubts;

1. Why take on $f_i =|G_{P_i}: H_{P_i}I_{G,P_i}|$, $H_{P_i}I_{G,P_i}$? and not only $H_{P_i}$? (p. 34)
2. If we choose a system of representatives $\{ \sigma_{ij} \}$ of $G_{P_1} \mod G_{P_1} \cap \tau_i H \tau_i^{-1}$. Why $\{ \sigma_{ij} \tau_i \}$ is a system of representatives on the left of $G\mod H$? (p. 35)
3. Why matrix B is taken in this way? (p. 36)

References: http://dc.library.okstate.edu/cdm/ref/collection/theses/id/2983 and Jürgen Neukirch- Algebraic number theory (p. 523)

Any help is appreciated!

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