# Lecture 23 (Sketch)

\todo[inline]{What is an isogeny?}
\todo[inline]{What is an Artin-Schreier extension?}
\todo[inline]{What is Kummer theory?}
\todo[inline]{What are Weil differentials?}
\todo[inline]{What are Kahler differentials?}
\todo[inline]{What is the Riemann Hurwitz formula?}

:::{.corollary title="?"}
Let $k$ be a perfect field of characteristic $p>0$, $d\in \ZZ^{\geq 0}$ with $\gcd(d, p) = 1$, and let $f\in k[x]$ and $L\da K(p^{-1} (f))$.
Then $[L:K] = p$ and $L/k$ is a regular function field of genus $g = {1\over 2}(p-1)(d-1)$ that is unramified away from $\infty$.
:::

## Artin-Schreier Extensions of Function Fields

:::{.fact}
For $k$ a field, $\ch(k) = p > 0$, and $a, b\in k$, TFAE:

1. $k(p^{-1}(a)) = k(p^{-1}(b))$

2. $a$ and $b$ generate the same cyclic subgroup of $k/p(k)$.

In particular, if $K(p^{-1}(u))/k$ is an Artin-Schreier extension, then for all $x\in k$, $k(p^{-1}(u - (x^p - x))) = k(p^{-1}(u))$.
:::

:::{.lemma title="?"}
Let $k$ a *perfect* field of characteristic $p>0$, $K/k$ a function field, $u\in K$, and $p\in \Sigma(K/k)$.

- There exists a $z\in K$ such that $z_v \da v_p(u - (z^p - z))$ satisfies either

  - $z_v \geq 0$, or
  - $z_v\leq 0$ and $z_p$ is prime to $p$.

- There exists a most one $m\in \ZZ$ that is negative and prime to $p$ such that for some $z\in K$ we have $v_p(u - (z^p - z)) = m$.
If such an $m$ exists, it is given by $m = \max\ts{v_p(u - (z^p - z)) \st z\in K }$.

- It follows that precisely *one* of the two alternatives in the first statement holds.
:::

:::{.theorem title="Genus Formula for Artin-Schreier Extensions"}
Let $k$ a *perfect* field of characteristic $p>0$, $K/k$ a function field, $u\in K$, $L\da K(p^{-1}(u))$, $p\in \Sigma(K/k)$, and set
\[  
M_p \da 
\begin{cases}
\abs{m} & \text{if there exists a $z\in K$ such that } v_p(u - (z^p - z)) = m \\
-1 & \text{if there exists a $z\in K$ such that } v_p(u - (z^p - z)) \geq 0 \\
\end{cases}
.\]

Then

a. If $M_p = -1$, then $p$ is unramified in $L$.

b. If $M_p \geq 1$, then $p$ is totally ramified in $L$.
  Letting $\tilde p$ be the unique place lying over $p$, then
  \[  
  d(\tilde p / p) = (p-1)(M_p + 1) && \text{(wild ramification)}
  .\]

c. Suppose there exists a $p$ such that $M_p \geq 1$.
  Then $[L: K] = p$, $L/k$ is regular, and we have a genus formula
  \[  
  g_L = p g_K + \qty{p-1 \over 2} \qty{ -2 + \sum_{p\in \Sigma(K/k)}(M_p+1)\deg p }
  .\]

:::