Examples of real places of a field K

Question

I am studying the basic theory of valuations and places of a field $K$. The definition of a place is a homomorphism $p$ : $\mathcal{O_v} \longrightarrow Kp$, where $\mathcal{O_v}$ is a valuation ring. This homomorphism must satisfy 1. if $x \not\in \mathcal{O_v}$, then $p(x^{-1})=0$ (note that $x^{-1}\in \mathcal{O_v}$ because $\mathcal{O_v}$ is a valuation ring) and 2. $p(x)\neq 0$ for some $x\in \mathcal{O_v}$ . We say that $Kp$ is the residue field of $p$. We can also define $p$ : $ K \longrightarrow Kp\cup\{\infty\}$, by setting $p(x)=\infty$ if $x\not\in \mathcal{O_v}$. I am particularly interested in $\mathbb{R}$- places, that is, a place $p$ : $K \longrightarrow \mathbb{R}\cup\{\infty\}$. Can anyone provide some bibliography where I can find good examples of the topic? Or even better, can you provide an example of a $\mathbb{R}$- place?
Thank you!

Answer 1

The thing is that $\mathbb{R}$ is absolutely not special in this perspective. You can construct places with residue field $F$ in standard fashions, and the fact that $F=\mathbb{R}$ does not bring any particular feature.

For instance, take $K=\mathbb{R}(x)$, the field of rational fractions. Then you have lots of places on $K$ with residue field $\mathbb{R}$: one for any $a\in \mathbb{R}$, given by the $(X-a)$-adic valuation. Explicitly, $p:\mathbb{R}(x)\to \mathbb{R}\cup \{\infty\}$ is then given by $R\mapsto R(a)$. You can see that this works for any field $F$ instead of $\mathbb{R}$.

As a good reference for the theory of places and valuations I like "Valuations, Orderings and Milnor K-Theory" by Efrat.

Answer 2

The notion of "places" is classical and well established in number theory, although there there may be some subtle disagreements among experts concerning the extensions of a place in an extension of number fields. A detailed introduction (with examples) can be found in G. Gras' book "Class Field Theory - From theory to practice", Springer LNM, chap. I, §§1-2. Concerning your specific question on real places, see a "digest" in https://math.stackexchange.com/a/3110703/300700 and https://math.stackexchange.com/a/3114286/300700