Kummer–Artin–Schreier–Witt theory

Definition

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

The Kummer part is Kummer theory. It applies to an integer $m \ge 1$ not divisible by the characteristic 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 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 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 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 $\operatorname{Gal}(K_\Delta/K) \cong \Hom(\Delta/\wp W_n(K), \Z/p^n\Z)$.