Stickelberger's theorem

In mathematics, Stickelberger's theorem is a result of algebraic number theory, which gives some information about the Galois module structure of class groups of cyclotomic fields. A special case was first proven by Ernst Kummer (1847) while the general result is due to Ludwig Stickelberger (1890).

Let $K m$ denote the $m$ th cyclotomic field, i.e. the extension of the rational numbers obtained by adjoining the $m$ th roots of unity to $ℚ$ (where $m \geq 2$ is an integer). It is a Galois extension of $ℚ$ with Galois group $G m$ isomorphic to the multiplicative group of integers modulo $m$ $(ℤ / m ℤ) \times$ . The Stickelberger element (of level $m$ or of $K m$ ) is an element in the group ring $ℚ [G m]$ and the Stickelberger ideal (of level $m$ or of $K m$ ) is an ideal in the group ring $ℤ [G m]$ . They are defined as follows. Let $ζ m$ denote a primitive $m$ th root of unity. The isomorphism from $(ℤ / m ℤ) \times$ to $G m$ is given by sending $a$ to $σ a$ defined by the relation

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