# Theorem of Stickelberger

In Mathematical, the theorem of Stickelberger is a result of the Algebraic theory of the numbers, which give certain information on the structure of the Module of Welshman of the groups of classes of the cyclotomic bodies. He is due to the Mathématicien Ludwig Stickelberger (1850 - 1936).

## Statement

That is to say $\ mathbb \left\{Q\right\} \left(\ zeta_m\right)$ a cyclotomic extension of body of $\ mathbb \left\{Q\right\}$ with a group of Welshman $G = \ \left\{\ sigma_a | has \ in \left(\ mathbb Z/m \ mathbb Z\right) ^* \\right\}$, and consider the group of ring $\ mathbb \left\{Q\right\}$. Let us define the element of Stickelberger $\ theta \ in \ mathbb \left\{Q\right\}$ by
$\ theta = \ frac 1 m \ sum_ \left\{1 \ has it \ the m, \left(has, m\right) =1\right\} has \ sigma_a^ \left\{- 1\right\}.$

and let us take $\ beta \ in \ mathbb \left\{Z\right\}$ such as $\ beta \ theta \ in \ mathbb \left\{Z\right\}$. Then $\ beta \ theta \,$ is an annihilator for the ideal group of class of $\ mathbb \left\{Q\right\} \left(\ zeta_m\right)$, like Module of Welshman.

Note that $\ theta \,$ itself does not need to be an annihilator, it is necessary simply that any multiple of this one in $\ mathbb \left\{Z\right\}$ is it.

## External bonds

• PlanetMath page

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