\input amstex
\input amsppt.sty

\nologo

\magnification1200
\hsize13cm
\vsize19cm

\def\Z{\Bbb Z}
\def\N{\Bbb N}
\def\al{\alpha}
\def\be{\beta}
\def\Om{\Omega}
\def\om{\omega}
\def\ph{\varphi}
\def\F{\Cal F}
\def\Nr{\operatorname{Nr}}

\topmatter 
\title The cyclotomic identity and the cyclic group
\endtitle 
\author A. Dress and Ch\. Siebeneicher
\endauthor 
\affil 
Universit\"at Bielefeld
\endaffil

\endtopmatter
\document

Metropolis and Rota \cite{1} discuss the cyclotomic identity
$$\frac {1} {1-\al t}=
\prod _{n\in\Bbb N} ^{}\left(\frac {1} {1-t^n}\right)^{M(\al,n)}.$$
Thereby they define the necklace-algebra $\Nr(A)$ for a commutative
ring $A$ with unit element 1. We show that $\Nr(A)$ is the
Burnside-ring $\hat\Om(\Z)$ of almost finite $\Z$-sets, where an
almost finite $\Z$-set is a set with an operation of the infinite
cyclic group $\Z$, such that every element lies in a finite orbit and
every orbit type $\Z/n\Z$ occurs only with finite multiplicity. For
every $n\in\N$ we have a homomorphism
$$\ph_n:\hat\Om(\Z)\longrightarrow\Z,$$
which assigns to a $\Z$-set the number of elements invariant under the
subgroup $n\Z$. The family $\ph_n$ with $n>0$ defines an injective
homomorphism 
$$\ph:\hat\Om(\Z)\longrightarrow \Z^J$$
of $\hat\Om(\Z)$ into the product ring $\Z^J$, where $J$ denotes the
set of positive integers.

By defining symmetric powers of almost finite $\Z$-sets, we get a
homomorphism 
$$s_t:\hat\Om(\Z)\longrightarrow 1+t\Om(\Z)[[t]]$$
of the additive group $\hat\Om(\Z)$ into the multiplicative group of
formal power-series with constant term 1 and coefficients in the ring
$\hat\Om(\Z)$. Combining this homomorphism with $\ph_1$, we get the
isomorphism 
$$\ph_1\circ s_t:\hat\Om(\Z)\longrightarrow 1+t\Z[[t]].$$
Finally we show that $\hat\Om(\Z)$ is isomorphic to the (generalized)
ring of Witt-vectors $W(\Z)$.

The Burnside-ring construction applies to every profinite group
providing thereby a generalized cyclotomic(?) identity. The
construction allows also to define for an arbitrary profinite group
and given commutative ring $A$ a generalized ring of Witt-vectors over
$A$.



\Refs
\ref\no1\by N. Metropolis and C.-C. Rota
\paper Witt vectors and the Algebra of Necklaces\jour
Adv\. in Math\.\vol 50\yr 1983\pages 95--125\endref


\endRefs





\enddocument
