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STICKELBERGER

Let $p$ be a prime, let $q=p^f$
Let 

Any prime ideal $\wp$ above $p$ in the ring Cyclotomic Ring offers a representation of the field ${\bf F}_q$

\begin{displaymath}{\bf F}_q\sim {\bf Z}[\zeta_p,\xi]/ \wp\end{displaymath}
 
The multiplicative character $\omega$ of ${\bf F}_q$ defines by
\begin{displaymath}\omega( class(\xi) ) = \bar\xi\end{displaymath}
has order $q-1$.
    [Stickelberger, 1890]
\begin{displaymath}- G_K(\omega^a) \equiv \frac{(\zeta_p-1)^{S(a)}}{R(a)}\mod (\zeta_p-1)^{S(a)}\wp\end{displaymath}
where $S(a)=\sum_{i=0}^{f-1} a_i$ and $R(a)=\prod_{i=0}^{f-1} a_i$. The $a_i$ are the$p$-digits of $a$.
$\circ$ divisibility of abelian codes
$\circ$ regular section groups
$\circ$ BWD-codes
$\circ$ Determination of Gauss sums



nextupprevious
Next:ILLUSTRATION Up:Les sommes de caractères Previous:ALMOST PERFECT SEQUENCES
Philippe Langevin