# Monday, November 09

## Chapter 1

Let $k$ be a field, not necessarily algebraically closed.

:::{.definition title="Algebraic Function Field"}
An one variable **algebraic function field** $F/K$ is a field extension $F$ of $K$ which factors as 

\begin{tikzcd}
F \ar[dr, "\text{algebraic}"] & \\
& k(x) \ar[dl, "\text{transcendental}"] \\
k
\end{tikzcd}

where $x\in \bar{k}$ is some element that is not algebraic over $k$.
:::

:::{.definition title="Field of Constants"}
The subfield
\[  
\tilde{k} \da \ts{z\in F \intersect K^{\text{alg}}} \leq F
,\]
consisting of elements that are algebraic over $F$ is denoted the **field of constants**.
:::

:::{.definition title="Algebraically Closed"}
If $\tilde k = k$, we say that $k$ is **algebraically closed** in $F$.
:::

:::{.definition title="Rational Function Field"}
An extension $F/k$  is **rational** iff $F = k(y)$ for some $y\in k^{\text{transc}}$ which is transcendental over $k$.
:::

:::{.definition title="Valuation Ring"}
A ring $\OO \subseteq F$ is a **valuation ring** for $F$ iff $k\subset \OO \subseteq  F$ and $z\in F \implies z\in \OO$ or $z^{-1} \in \OO$.
:::


:::{.definition title="Discrete Valuation Ring"}
A ring local $R$ (thus with a unique maximal ideal) which is a PID but not a field is a **discrete valuation ring**.
:::


:::{.definition title="Place"}
A **place** of a function field $F/K$ is the maximal ideal of a valuation ring of $F/K$.
:::


:::{.definition title="Discrete Valuation"}
A **discrete valuation** of $F/k$ is a function
\[  
v: F\to \ZZ\union\ts{\infty}
\]
that is

1. Nondegenerate: $v(x) = \infty$ iff $x=0$.
2. Multiplicative: $v(xy) = v(x) + v(y)$.
3. Ultrametric triangle inequality: $v(x+y) \geq \min(v(x), v(y))$.
4. Fiber over one: there exist a $z\in F$ with $v(z) = 1$.
5. $\ro{v}{k} = 0$.
:::


:::{.definition title="Rational Place"}
A place of degree one is said to be a **rational place**.
:::


:::{.definition title="Valuation Ring of a Place"}
The **valuation ring of a place** is defined by
\[  
\OO_p \da \ts{z\in F \st z^{-1} \not\in P}
.\]
:::

:::{.definition title="Degree of a Place"}
The **degree** of a place $P$ is defined by
\[  
\deg(P) \da [F_p : k]
,\]
where $F_p = \OO_P / P$.
:::

:::{.definition title="Discrete Valuation of a Place"}
To any place $P$ we associate the function
\[  
v_p: F\to \ZZ\union\ts{\infty}
\]
defined by choosing any prime $t\in P$, writing any $x\in F$ as $x = t^n u$ with $u \in \OO_P\units$, and setting
\[  
v_p(x) = 
\begin{cases}
n & \text{if } x = t^n u \\ 
0 & x=\infty.
\end{cases}
\]

:::

Note: from now on we assume $\tilde K = K$


:::{.definition title="Divisor"}
The **divisor group** of $F/K$ is the free abelian group on the set of places of $F/K$, i.e. a formal sum
\[  
D = \sum_{\text{Places} p} b_p P \qquad n_p \in \ZZ
m\]
where cofinitely many $n_p$ are zero.
:::

:::{.definition title="Degree of a Divisor"}
The **degree** of a divisor is defined by 
\[  
\deg(D) \da \sum_P v_p(D) \deg(P)
.\]
:::

:::{.definition title="Principle Divisors"}
The set of divisors
\[  
\operatorname{Princ}(F) \da \ts{(x) \st 0\neq x \in F}
.\]
:::

:::{.definition title="Divisor Class Group"}
TRh
:::

:::{.definition title="Riemann-Roch Space"}
For a divisor $A\in \div(F)$, the **Riemann-Roch** space is defined as

\[  
\mathcal{L}(A) \da \ts{x\in F \st (x) \geq - A}\union \ts{0}
.\]

:::