STICKELBERGER

Let $p$ be a prime, let $q=p^f$
Let 

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

$${\bf F}_q\sim {\bf Z}[\zeta_p,\xi]/ \wp$$

 
The multiplicative character $\omega$ of ${\bf F}_q$ defines by

$$\omega( class(\xi) ) = \bar\xi$$

has order $q-1$.

    [Stickelberger, 1890]

$$- G_K(\omega^a) \equiv \frac{(\zeta_p-1)^{S(a)}}{R(a)}\mod (\zeta_p-1)^{S(a)}\wp$$

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