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 $L=k(\root n\of D)$ for some order-$n$ cyclic subgroup $D\subset k^\times/k^{\times n}$, and conversely.

(Something can be said even when $\zeta\notin k$ 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 $D\subset k/\wp(k)$, where $\wp:k\to k$ is the endomorphism $x\mapsto x^p-x$ 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 $D\subset W_m(k)/\wp(W_m(k))$, 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 $K(\root p^m\of\alpha)|K$ 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.