Places of this extension

Question

I'm reading this book.

I'm trying to find the degree of the places of the extension $\mathbb C(X)\mid\mathbb R$.

I know the places of the extension $\mathbb R(X)\mid\mathbb R$ and I've already proved that the places of the former are in the latter, but I don't if there are more places in $\mathbb C(X)\mid\mathbb R$.

I need help.

Thanks in advance.

Answer 1

According to Stichtenoth a place $P$ of $\mathbb{C}(X)|\mathbb{R}$ is the maximal ideal of a valuation ring of $\mathbb{C}(X)$ containing $\mathbb{R}$.

Claim: such a valuation ring contains $\mathbb{C}$.

Proof: let $O$ be a valuation ring of $\mathbb{C}(X)$ containing $\mathbb{R}$. Then either $i$ or $i^{-1}=-i$ are in $O$. Thus $\mathbb{C}=\mathbb{R}[i]=\mathbb{R}[-i]\subset O$, since $O$ is a ring.

We conclude that every place of $\mathbb{C}(X)|\mathbb{R}$ is a place of $\mathbb{C}(X)|\mathbb{C}$. The latter are the maximal ideals of the form $(X-z)\mathbb{C}[X]_{(X-Z)}$, $z\in\mathbb{C}$ and $X^{-1}\mathbb{C}[X^{-1}]_{(X^{-1})}$.

It is not true, that every place of $\mathbb{R}(X)|\mathbb{R}$ yields exactly one place of $\mathbb{C}(X)|\mathbb{R}$: for example $(X^2+1)\mathbb{R}[X]_{(X^2+1)}$ is a place of $\mathbb{R}(X)|\mathbb{R}$. It extends however to two different places of $\mathbb{C}(X)|\mathbb{R}$, namely $(X-i)\mathbb{C}[X]_{(X-i)}$ and $(X+i)\mathbb{C}[X]_{(X+i)}$.