I am studying the proof of Proposition 1 presented in this paper. I am stuck at the following highlighted lines. Can anyone please help me understand the math behind them? Any help would be appreciated.
Also, when posting this question, I could not give an English description of $(\mathcal{O}_K/p\mathcal{O}_K)^\star$. How do I describe that unit group in English for the title or when I am presenting it in a group discussion?
