\classheader{2013-03-04}

We mentioned last time that there was a correspondence between algebra and geometry,
\[\begin{array}{c}
\text{function fields in one}\\
\text{variable over }\C
\end{array}\longleftrightarrow\begin{array}{c}
\text{compact Riemann surfaces}
\end{array}\]
You can find some good books about this in the Eckhart library, in section QA333. One good book is G. Spring's \textit{Introduction to Riemann Surfaces}.

An algebraic formulation of this correspondence is
\[K\longleftrightarrow X,\]
where
\[X=\{\text{discrete valuation rings }V\mid k\subset V\subset K\text{ and }\mathrm{Frac}(V)=K\}.\]
A discrete valuation ring is a regular local noetherian ring of dimension one. Equivalently, we could define it to be a local ring whcih is a PID, but not a field. Some examples are $\C[T]_{(T)}$, and $\Z_{(p)}$.

An example of how to think about this correspondence is if
\[X=\Spec(A)\mathbin{\underset{\Spec(A'')}{\cup}}\Spec(A').\]
Thus, $x\in X$ equals either a maximal ideal $\frak{m}\in\max(A)$, or $\frak{m}\in\max(A')$. Letting $V=A_\frak{m}$ or $A'_\frak{m}$ as the case may be, this is a DVR, and then the point $x$ corresponds to this $V$, and $X$ can be understood as the set of such $V$.

If $K$ is a function field in one variable over $\C$, we can let
\[V=\{f\in K\mid f\text{ is holomorphic at }x\}.\]
Then $V$ is a PID, with a prime element $t$ (which generates the unique maximal ideal of $V$) given by a function which vanishes with order 1 at $x$. Thus, any $f\in K^\times$ can be written uniquely as $f=ut^n$ where $u\in V^\times$, and $n$ is the order of the zero / pole of $f$ at $x$.

Let's consider the example of $K=\C(T)(\sqrt{f(T)})$ where $f(T)=(T-\alpha_1)\cdots(T-\alpha_n)$, the $\alpha_i$ being distinct. Assume also that $n$ is odd.