Take $K/\mathbb{Q}$ a number field with ring of integers $\mathcal{O}_K$. Then an ideal $I \subset \mathcal{O}_K$ is a free $\mathbb{Z}$-module of dimension $N = [K:\mathbb{Q}]$ with basis $(\alpha_1,\ldots, \alpha_N)$, that is : $$I = \left\{ \sum_{n=1}^N m_n \alpha_n, m_n \in \mathbb{Z}\right\}$$ Then we have $N$ complex embedding $\sigma_n : K \to \mathbb{C}$ (the Galois group of the normal closure) and we have the linear operator $\mathbb{C}^N \to \mathbb{C}^N$ : $$T \quad :\quad \begin{pmatrix} m_1\\ \vdots \\ m_N \end{pmatrix} \mapsto \begin{pmatrix} \sum_{n=1}^N m_n \sigma_1(\alpha_n)\\ \vdots \\ \sum_{n=1}^N m_n \sigma_N(\alpha_n) \end{pmatrix}$$ An important theorem is that $T^{-1}$ is of the form $$T^{-1} \quad :\quad \begin{pmatrix} m_1\\ \vdots \\ m_N \end{pmatrix} \mapsto \begin{pmatrix} \sum_{n=1}^N m_n \sigma_1(\frac{\beta_n}{\delta})\\ \vdots \\ \sum_{n=1}^N m_n \sigma_N(\frac{\beta_n}{\delta}) \end{pmatrix}$$ $$\color{red}{\text{where }(\beta_1,\ldots,\beta_N)\text{ is a basis for the ideal }J \subset \mathcal{O}_K\text{ such that }IJ = (\delta)}$$
This is from this theorem that we get the functional equation for the Dedekind zeta function $\zeta_K(s)$.
It is proven in Hecke p.126 and in Neukrich p.218 but it is hard to follow their proof. In those books, for some $c \in K$, the notation for $\sigma_p(c)$ is $c^{(p)}$.
Can you help understanding the main argument ?
If it is easier, we can look first at $K$ totally real, so that $\sigma_n$ are real embeddings and $T$ is a linear operator $\mathbb{R} \to \mathbb{R}$.