nt.number theory - What is interesting/useful about big Witt Vectors?

I am particularly interested in knowing their original motivation and applications.

Kummer theory says that every degree-$n$ cyclic extension $L|k$ of any field $k$ containing a primitive $n$-th root $zeta$ of $1$ is of the form $L=k(root nof D)$ for some order-$n$ cyclic subgroup $Dsubset k^times/k^{times n}$, and conversely.

(Something can be said even when $zetanotin k$ but when $n$ is invertible in $k$. Look up a certain exercise in Schoof's book on Catalan's Conjecture.)

This leaves out degree-$p$ cyclic extensions $L|k$ of a characteristic-$p$ field. Artin-Schreier proved that $L=k(wp^{-1}(D))$ for some ${mathbb F}_p$-line $Dsubset k/wp(k)$, where $wp:kto k$ is the endomorphism $xmapsto x^p-x$ of the additive group $k$, and conversely.

What about degree-$p^m$ cyclic extensions $L|k$ of a characteristic-$p$ field ? Many complicated constructions for particular cases were given in the 1930s (by people such as Albert) before Witt introduced the ring $W_m(k)$ of $p$-typical Witt vectors of length $m$ and the endomorphism $wp:W_m(k)to W_m(k)$ of the additive group, and proved that $L=k(wp^{-1}(D))$ for some order-$p^m$ cyclic subgroup $Dsubset W_m(k)/wp(W_m(k))$, and conversely.

There were many other papers in the same volume of Crelle 176 (1937) applying Witt vectors to other outstanding problems. My favourite is Hasse's characterisation of those $alpha$ in a finite extension $K$ of ${mathbb Q}_p$ containing a primitive $p^m$-th root of $1$ for which the extension $K(root p^mofalpha)|K$ is unramified ($p^m$-primary numbers; see for example the book by Fesenko and Vostokov, freely available on Fesenko's homepage).

See also Harder, Wittvektoren,
Jahresber. Deutsch. Math.-Verein. 99 (1997), no. 1, 18--48.

An English translation of this paper has appeared in Ernst Witt, Gesammelte Abhandlungen, Springer, Berlin, 1996.