The way conjugation acts on embeddings

Question

Let $K$ be a field, $F$ be complex conjugation, and let $K_{\mathbb{C}} = \prod_{\tau} \mathbb{C}$, where $\tau$ is taken over the set of embedding $K \hookrightarrow \mathbb{C}$. Then for $z = (z_{\tau}) \in K_{\mathbb{C}}$, $Fz_{\tau} = \overline{z_{\overline{\tau}}}$. My question is, why do both $z$ and $\tau$ get conjugated. It seems like only one or the other should be conjugated.

Edit: I've come to the realization that conjugating both $z$ and $\tau$ is just the way that the action of $F$ is defined, but I'm still a curious as to why.

Answer 1

Somewhat more canonically, $K_{\mathbb C} = \mathbb C\otimes_{\mathbb Q} K$. Complex conjugation then acts just by its action on the first factor in the tensor product.

This tensor product is a product of copies of $\mathbb C$, labelled by the embeddings $\tau$; the projection onto the $\tau$th factor is then just the embedding $\tau$. The isomorphism is given by $$ z\otimes x \mapsto ( z \tau(x) )_{\tau}$$ (where $z \in \mathbb C$ and $x \in K$).

Concretely, then, conjugation take a tuple $(z_{\tau})_{\tau}$ to the tuple $(\overline{z}_{\overline{\tau}})_{\tau}$, just because $$z\otimes x \mapsto \overline{z} \otimes x \quad \text{(this is complex conjugation on the tensor product) }$$ $$\mapsto \bigl( \overline{z} \tau(x) \bigr)_{\tau} = \bigl(\overline{z\, \overline{\tau}(x)} \bigr)_{\tau}$$ (using the formula $\overline{\tau(x)} = \overline{\tau}(x),$ which can be rewritten as $\tau(x) = \overline{\overline{\tau}(x)}$).

Summary: the complex conjugation is the obvious, natural, one if you think of $K_{\mathbb C}$ as $\mathbb C\otimes_{\mathbb Q} K.$

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As Alex points out in his answer, the fixed part of complex conjugation is then $\mathbb R\otimes_{\mathbb Q} K$, which can be described (as in his answer) as the product of $r_1$ copies of $\mathbb R$ and $r_2$ copies of $\mathbb C$.

Answer 2

The reason is that, for many purposes, a more convenient space to embed $K$ into is $$ \prod_v \mathbb{R} \times \prod_{\tau/\sim}\mathbb{C},\;\;\;\;\;\;\;\;\;\;\;(*) $$ where the first product runs over the real embeddings, and the second product is taken over representatives of complex conjugate pairs of embeddings. The idea is that you do not need to keep track of $\bar{\tau}$, once you know $\tau$, since it does not carry any new information.

Now, the way your complex conjugation operator $F$ is defined, its fixed space in $\prod_{\tau}\mathbb{C}$ can be identified with the product (*) (to be precise, it consists of tuples with arbitrary real values at the real embeddings, and with the entry at $\tau$ being the complex conjugate of the entry at $\bar{\tau}$ for any complex embedding $\tau$), while the image of $K$ in $\prod_\tau\mathbb{C}$ is fixed by $F$. So you can simply obtain the desired embedding into (*) by restricting the embedding you have written down to the fixed space under $F$.

If you had defined $F$ in the "naive" way, you wouldn't be able to just restrict the embedding you have written down to the fixed subspace of $F$ and hope to obtain anything sensible, because you would have lost a huge chunk of $K$ this way. You would have only retained the totally real elements of $K$, i.e. those that land in $\mathbb{R}$ under any complex embedding of $K$.