---
tag: SVHTAFD07A
kind: definition
name: Kummer–Artin–Schreier–Witt theory
msc: [12F10, 13F35, 12G05]
defines: R45FHBE15X
uses: [0QA8CHH8MG, 1ZB13N78XW, 6N3X2QX4W9, JJ91A1V8NQ, EQDEXBWGJH, 0XTY86BZKY, 0T3WVTD675, XZQENYV15Y, 7YNMNRGH3T]
sources:
  - url: https://en.wikipedia.org/wiki/Witt_vector
---

## Definition

Let $K$ be a [[field|0QA8CHH8MG]] and let $K^s$ be a [[separable closure|JJ91A1V8NQ]] of $K$.
*Kummer–Artin–Schreier–Witt theory* is the description of the finite [[abelian extensions|EQDEXBWGJH]] of $K$ inside $K^s$ of exponent dividing a given integer.
It has two parts.

The Kummer part is [[Kummer theory|XZQENYV15Y]].
It applies to an integer $m \ge 1$ not divisible by the [[characteristic|1ZB13N78XW]] of $K$, when $K$ contains $m$ distinct $m$-th roots of unity.
It describes the abelian extensions of exponent dividing $m$ by adjoining $m$-th roots.

The Artin–Schreier–Witt part concerns a field $K$ of characteristic $p > 0$ and an integer $n \ge 1$.
Let $W_n(K)$ be the ring of [[Witt vectors|6N3X2QX4W9]] of length $n$.
Its elements are sequences $(a_0, \ldots, a_{n-1})$ in $K$, with the first $n$ Witt sum and product polynomials as operations.
Let $F(a_0, \ldots, a_{n-1}) = (a_0^p, \ldots, a_{n-1}^p)$ be the [[Frobenius|0T3WVTD675]] endomorphism of $W_n(K)$, and let $\wp = F - \id$, an additive homomorphism.
For a group $\Delta$ with $\wp W_n(K) \subseteq \Delta \subseteq W_n(K)$, let $K_\Delta$ be the field generated over $K$ by the entries of all $x \in W_n(K^s)$ with $\wp(x) \in \Delta$.
The theory states:

1. the map $\wp \colon W_n(K^s) \to W_n(K^s)$ is surjective, and its kernel is $W_n(\F_p) \cong \Z/p^n\Z$;
2. the map $\Delta \mapsto K_\Delta$ is a bijection from the groups $\Delta$ as above with $\Delta / \wp W_n(K)$ finite onto a set of extensions.
   This set consists of the finite abelian extensions of $K$ inside $K^s$ whose [[Galois group|0XTY86BZKY]] has exponent dividing $p^n$;
3. for such $\Delta$, the pairing $\operatorname{Gal}(K_\Delta/K) \times \Delta/\wp W_n(K) \to \Z/p^n\Z$, $(\sigma, a) \mapsto \sigma(x) - x$ with $\wp(x) = a$, is well defined and perfect, so $\operatorname{Gal}(K_\Delta/K) \cong \Hom(\Delta/\wp W_n(K), \Z/p^n\Z)$.

## Remarks

For $n = 1$, $W_1(K) = K$ and $\wp(x) = x^p - x$, so the Artin–Schreier–Witt part is [[Artin–Schreier theory|7YNMNRGH3T]].
The cyclic extensions of degree $p^n$ correspond to the groups $\Delta$ generated by $\wp W_n(K)$ and one $a$ whose class has order $p^n$.

Example: take $K = \F_p$ and $n = 2$.
Here $F$ is the identity on $W_2(\F_p)$, because $c^p = c$ for $c \in \F_p$, so $\wp W_2(\F_p) = 0$.
Thus $W_2(\F_p)/\wp W_2(\F_p) \cong \Z/p^2\Z$ is cyclic of order $p^2$.
By clauses 2 and 3, $\F_p$ has exactly one extension in $K^s$ with Galois group of exponent dividing $p^2$ and of degree $p^2$, and it is cyclic: it is $\F_{p^{p^2}}$.

Non-example: the Kummer part needs the roots of unity in the base field.
Over $\Q$ with $m = 3$, the field $\Q(\sqrt[3]{2})$ is not Galois over $\Q$, since its other conjugates $\omega \sqrt[3]{2}$, with $\omega$ a primitive cube root of unity, are not real.
So adjoining $m$-th roots does not give abelian extensions when $K$ lacks the $m$-th roots of unity.