I am reading the book Introduction to Cyclotomic Fields by Lawrence C. Washington and I am stuck in proving the relation $\varepsilon_i \theta = \frac{1}{p} \sum_{a=1}^{p-1} a{\omega}^{-1}(a)\varepsilon_i$ (the last relation on the attached image). It is mentioned that we have to use the identity (d) $\varepsilon_\chi \sigma = \chi(\sigma)\varepsilon_\chi$ somehow.
My approach:
$\varepsilon_i \theta = \frac{1}{p-1} \sum_{a=1}^{p-1} a{\omega}^{i}(a) \sigma(a^{-1}) \cdot \frac{1}{p} \sum_{a=1}^{p-1} a{\sigma_a}^{-1}$ (From the definition of $\varepsilon_i$ and $\theta$)
$= \cdots$
$= \cdots$
$= \frac{1}{p} \sum_{a=1}^{p-1} a{\omega}^{-i}(a)\varepsilon_i $
$=B_{1,\omega^{-1}} \varepsilon_i $
How do I use (d) to fill the gaps in the proof? Please, help me.
