Neukirch Definition of Map into $K_{\mathbb{C}}$ I'm confused about how Neukirch defines this $j$ map at the beginning of his section on Minkowski theory. Here is his definition:

I don't really understand what $K_{\mathbb{C}}=\prod_{\tau}\mathbb{C}$ means. I understand that $\tau$ must range over the $n$ complex embeddings of $K$ into $\mathbb{C}$, but beyond that I'm pretty lost. Also, what does it mean for $a$ to map to $ja$? What is $j$ here, and why does he say this is equal to $(\tau a)$?
I really appreciate any help.
 A: If Neukirch uses standard notations (which I assume he will), the ring $\prod_{\tau} \mathbb C$ is a finite product of copies of $\mathbb C$, one copy for each embedding $\tau : K \to \mathbb C$. So as a $\mathbb C$-vector space, it is isomorphic to $\mathbb C^n$, but we also give it a ring structure (by multiplying component-wise). Say the embeddings of $K$ are given by the set $\{\tau_1,\cdots,\tau_n\}$. The embedding $j : K \to K_{\mathbb C} = \prod_{\tau} \mathbb C$ sends an element $a \in K$ to the $n$-tuple $j(a) \overset{def}= (\tau_1(a), \cdots, \tau_n(a))$. At least that is what I am reading from the text you copied above. 
Added : given an element $x \in \prod_{\tau} \mathbb C$, Neukirch denotes its $\tau$-component (i.e. the $i^{\text{th}}$ component if $\tau = \tau_i$) by $x_{\tau}$. So the notation
$$
\langle x,y \rangle \overset{def}= \sum_{\tau} x_{\tau} \overline{y_{\tau}}
$$
can be understood in standard vector notation as 
$$
\langle (x_1,\cdots,x_n),(y_1,\cdots,y_n) \rangle \overset{def}= \sum_{i=1}^n x_i \overline{y_i}
$$
where the $i^{\text{th}}$ component corresponds to $\tau_i$ in the embedding $j : K \to K_{\mathbb C}$ defined by $j(a) = (\tau_1(a),\cdots,\tau_n(a))$. In particular, $\|j(a)\|^2 = \langle j(a),j(a) \rangle = \sum_{\tau} |\tau(a)|^2$, where the sum ranges over all embeddings $\tau : K \to \mathbb C$. 
Hope that helps,
A: $\prod_{\tau}\mathbb{C}$ is the product of $n$ copies of $\mathbb{C}$, one copy for each complex embedding $\tau$ of $K$ into $\mathbb{C}$.
$ja=(\tau a)$ means that $j(a)$ is the $n$-tuple formed by the values of each $\tau$ on $a$.
Write $j(a)=(\tau_1(a), \dots, \tau_n(a))$ if it feels more familiar.
