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I am working through "Algebraic function fields and codes-Henning Stichtenoth" (2nd Edition) and I got a little confused about the following:

First of all an algebraic function field $F/K$ of one variable over $K$ is an extension field $F$ of $K$ such that $F$ is a finite algebraic extension of $K(x)$ for some element $x \in F$ which is transcendental over $K$.

Remark 1.11.17 says that if $K$ is an algebraic closed field, then all places are rational ( A Place of $F/K$ is defined as the unique maximal ideal of a valuation ring $\mathcal{O}$ of $F/K$). So far so good, but isn't the following also true:

If there exists at least one rational place, then $K$ is algebraically closed in $F$(!), therefore all places are rational.

Because: The field of constants $\tilde K :=\{z \in F:z \text{ is algebraic over }K\}$ of $ F/K$ is embedded in $F_p:=\mathcal{O}_p/P$ (The residue class field of a Place $P$). So it follows that $[\tilde K:K]\leq[F_p:K]$. This holds for any Place $P$ of $F/K$. (part of proof of Corollary 1.1.16)

Therefore if there exists a Place $P$, which is rational (defined by deg$P=[F_p:K]=1$). Then $[\tilde K:K]\leq[F_p:K]=1$, therefore $K$ is already algebraically closed in $F$.

Then for any other Place $\hat P$, since deg$\hat P=[F_{\hat p}:K]$ has to be finite (Proposition 1.1.15), $F_{\hat p}:K$ is an algebraic extensions of $K$, which uses only elements of $F$, so it can't get any bigger than $K$. Therefore $\hat P$ is rational.

This strikes me odd, since there seems to be only two possibilties:
1) Either no Place $P$ of $F/K$ is rational,
2) or every Place $P$ of $F/K$ is rational(which is the case if $K$ is algebraic closed or algebraically closed in $F$).

Am I missing something? If yes could you please give me a hint where I went wrong.

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  • $\begingroup$ what do you mean by "place", in this mathematical context? $\endgroup$ Jul 20, 2017 at 9:41
  • $\begingroup$ Place of $F/K$ is defined in this context as the maximal Ideal of a valuation ring $\mathcal{O}$ of $F/K$. Edited the Question. $\endgroup$
    – FeLix
    Jul 20, 2017 at 10:11
  • $\begingroup$ A place of $F/K$ should not be the valuation ring but the maximal ideal of a valuation ring of $F/K$. I am only interested in the case of function fields, i am sorry if haven't made that clear enough. $\endgroup$
    – FeLix
    Jul 21, 2017 at 21:08
  • $\begingroup$ It is the same. So you meant a place of $F/K$ is a discrete valuation ring $\mathcal{O}$ (integral domain with a unique maximal ideal $P$) with $F = Frac(\mathcal{O})$ and $K \subset \mathcal{O}$. Then $K = K / (P \cap K) \subset \mathcal{O}/P$ and hence $K$ embeds in $\mathcal{O}/P$. We can then define the degree of the place as $[\mathcal{O}/P:K]$. Is it always finite ? $\endgroup$
    – reuns
    Jul 21, 2017 at 21:15
  • $\begingroup$ Yes, contained in $F$ and containing $K$. I am not sure if i understand you correctly. $K$ embeds in $\mathcal{O}/P$ but the elements are not the same, most importantly, an element of $ \mathcal{O}/P$ is not an element of $F$. Yes it always is finite. $\endgroup$
    – FeLix
    Jul 21, 2017 at 21:17

1 Answer 1

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I think I found my mistake. It is in the statement "$F_p$ is an algebraic extensions of $K$, which uses only elements of $F$, so it can't get any bigger than $K$." Because $K$ is only embedded in $\mathcal{O}_P/P=F_p$, it is not an actual subset. So the statement actually makes no real sense. Therefore I am right about the question I asked, but not on my further statement, that if there is one rational place, there can only be rational places.

For example, for the rational function field $K(x)$ a valuation ring is given by: \begin{align} \mathcal{O}_{p(x)}= \Big\{ \frac{f(x)}{g(x)}| f(x),g(x)\in K[x], p(x) \text{ does not divide }g(x) \Big\} \end{align} for an irreducible monic polynomial $p(x)\in K[x]$, the place associated to that valuation ring is: \begin{align} P_{p(x)}= \Big\{ \frac{f(x)}{g(x)}| f(x),g(x)\in K[x],p(x) \text{ does divide }f(x), p(x) \text{ does not divide }g(x) \Big\} \end{align}

The degree of a Place $P$ is defined by deg$P=[\mathcal{O}_{p(x)}/P:K]$, and $\mathcal{O}_{p(x)}/P$ is isomorphic to $K[x]/(p(x))$, with an isomorphism given by: $f(x)+(p(x)) \mapsto f(x)+P$. Therefore if $K$ is algebraically closed, every irreducible polynomial has degree one, therefore every place has degree one.

If $K=\mathbb{R}$ on the other hand take $p(x)=x^2+1$ and you get a place with degree $2$. Note that $\mathbb{R}$ is algebraically closed in $\mathbb{R}(x)$, this means $\tilde K=\mathbb{R}$.
If you look at $\mathcal{O}_{x^2+1}/P$, you see it contains more algebraic Elements over $\mathbb{R}$ than $\mathbb{R}$ itself does ($x^2+P$), but this is not a problem because $\tilde K$ is defined as $\{z \in F: z \text{ is algebraic over } K\}$, so in this case: \begin{align} \tilde K=\{z \in \mathbb{R}(x): z \text{ is algebraic over } \mathbb{R}\}=\mathbb{R} \end{align} and $x^2+P \in \mathcal{O}_{x^2+1}/P$ but on the other hand $x^2+P \not \in \mathbb{R}(x)$.

To sum things up; the conclusion I drew is partly wrong, if there is a rational place there still can be other places with a higher degree than $1$, although $K$ has to be algebraically closed in $F$.

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