Abelian number fields (1/2)
By Kronecker-Weber, these are subfields of cyclotomic fields. A good description for such a field
F is by a pair (f, H), where H is the subgroup of (Z/fZ)∗
= Gal(Q(ζf )/Q) fixing F. If f
is minimal, we call it the conductor of the extension: this is the critical parameter for all
complexities involved.
In PARI terms, we will use an argument fH to denote either of
an integer f, describing Q(ζf ) (implicitly H = (1));
a pair [f, H], where f is an integer and H is a vector of generators as t_INTMODs modulo
f or t_INTs (implicitly mapped to (Z/fZ)∗);
a pair [G, H] where G is idealstar(f, 1) and H is a subgroup, given by the canonical
HNF matrix giving the generators of H in terms of G.gen. This HNF matrix divides the
diagonal matrix with diagonal G.cyc and there is a one-to-one correspondence between
subgroups of a finite abelian group and such matrices; the determinant of the matrix is equal
to the subgroup index.
Atelier 2022 (13/01/2022) – p. 3/13