How is the Minkowski space $K_{\Bbb R}$ and $j: K\to K_{\Bbb R}$ defined? (Neukirch)

Question

The basic setup is as follows. Let $K$ be a number field of degree $n$, define $K_{\Bbb C} = \prod_{\tau} \mathbb C$ and consider the canonical mapping $j: K\to K_{\Bbb C}$ given by $j(a) = (\tau a)_\tau$, i.e., if $\tau_1,\ldots,\tau_n: K \hookrightarrow \Bbb C$ are the $n$ embeddings of $K$, then $j(a) = (\tau_1(a), \ldots, \tau_n(a))$. Next, consider the Galois group $G(\Bbb C/\Bbb R)$ generated by complex conjugation $F: z\to \bar z$. To each embedding $\tau: K \hookrightarrow \Bbb C$, there exists a conjugate embedding $\overline{\tau}: K \hookrightarrow \Bbb C$ obtained by $\overline{\tau} := F \circ \tau$. Altogether, this defines an involution $F: K_{\Bbb C} \to K_{\Bbb C}$ (note the abuse of notation), which is given by $(Fz)_\tau = \overline{z}_{\overline{\tau}}$ on points $z = (z_\tau) \in K_{\Bbb C}$.

At this stage, the author defines $K_{\Bbb R}$ to be the $G(\Bbb C/\Bbb R)$ invariant, i.e., $F$-invariant points of $K_{\Bbb C}$. The notation is $K_{\Bbb R} = K_{\Bbb C}^+$. The book says:

These are the points $(z_\tau)$ such that $z_{\overline\tau} = \overline{z}_\tau$. An explicit description of $K_{\Bbb R}$ will be given anon. Since $\overline{\tau} a = \overline{\tau a}$ for every $a\in K$, one has $F(ja) = ja$. This yields a mapping $$j:K\to K_\Bbb R$$

1. Could someone explain what $K_{\Bbb R}$ is in greater detail? My understanding is that $G(\Bbb C/\Bbb R)$ consists of automorphisms $\phi: \Bbb C\to \Bbb C$ fixing $\Bbb R$ pointwise, and there are only two options, i.e., $\phi_1(z) = z$ and $\phi_2(z) = \overline z$. $\phi_2$ is just $F$, so to consider $G(\Bbb C/\Bbb R)$ invariant points of $K_{\Bbb C}$, it is enough to consider $F$-invariant points of $K_{\Bbb C}$. Suppose a point $z = (z_\tau) \in K_{\Bbb C}$ is $F$-invariant, i.e., $Fz = z$. Then, coordinate-wise, we have $(Fz)_\tau = \overline{z}_\overline\tau = z_\tau$. Since this holds for every $\tau$, we can replace $\tau$ by $\overline \tau$ to get $z_{\overline\tau} = \overline{z}_\tau$ as the author claims, but I do not see the reason behind this odd presentation.

2. It's clear that $\overline \tau a = \overline{\tau a}$ for all $a\in K$, since $\overline{\tau} a = F \circ \tau (a) = \overline{\tau a}$. Why is $F(ja) = ja$? Let $a\in K$, then $ja = (\tau a)_\tau$. Using the definition of $F$ above, $$(F(ja))_\tau = \overline{ja}_\overline\tau = (\overline{\tau a})_{\overline\tau} = (\overline\tau a)_{\overline\tau}$$ but this doesn't seem to do it since $ja = (\tau a)_\tau$ and not $(\overline\tau a)_{\overline\tau}$. Am I missing something trivial?

3. How is the map $j: K\to K_\Bbb R$ constructed? Perhaps this will depend on $F(ja) = ja$.

Thanks a lot!

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Reference: Neukirch's Algebraic Number Theory.

Answer 1

1. I don't think there is any greater detail to be given. The points in $K_{\mathbb{C}}$ are indexed by embeddings $\tau$. There are two type of embeddings: the real ones, with image in $\mathbb{R}$, and the non-real ones, which go by pairs of conjugates. Then $(z_{\tau})_{\tau}$ is in $K_{\mathbb{R}}$ if for each real embedding $\tau$, the coordinate $z_{\tau}$ is a real number, and for each pair $\{\tau,\bar{\tau}\}$ of conjugate embeddings, the corresponding coordinates $z_{\tau}$ and $z_{\bar{\tau}}$ are conjugate complex numbers. That's it. There is nothing deeper.

2. You missed that $(\tau a)_{\tau}$ and $(\bar{\tau} a)_{\bar{\tau}}$ are litteraly the same thing by definition. This is just a reindexing.

3. The map $j: K\to K_{\mathbb{R}}$ is just the map $j: K\to K_{\mathbb{C}}$. The fact that $F(ja)=ja$ means that the image of $j: K\to K_{\mathbb{C}}$ is included in $K_{\mathbb{R}}$, so $j$ is actually a map from $K$ to $K_{\mathbb{R}}$.