Minus class groups of the fields of the 𝑙-th roots of unity

Schoof, René

doi:10.1090/S0025-5718-98-00939-9

Minus class groups of the fields of the $l$-th roots of unity
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Math. Comp. 67 (1998), 1225-1245 Request permission

Abstract:

We show that for any prime number $l>2$ the minus class group of the field of the $l$-th roots of unity $\overline {\mathbf {Q}_p} (\zeta _l)$ admits a finite free resolution of length 1 as a module over the ring $\widehat {\mathbf {Z}} [G]/(1+\iota )$. Here $\iota$ denotes complex conjugation in $G={{Gal}}( \overline {\mathbf {Q}_p} (\zeta _l)/\overline {\mathbf {Q}_p} )\cong (\mathbf {Z} /l\mathbf {Z} )^*$. Moreover, for the primes $l\le 509$ we show that the minus class group is cyclic as a module over this ring. For these primes we also determine the structure of the minus class group.

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