Abelian number fields (2/2)
a pair [G, H] where G is a bnr structure attached to a ray class group Clf(Q) and H is a
subgroup given by a canonical HNF matrix; the place at infinity must not be forgotten: after Q
= bnfinit(y), the structure bnrinit(Q, [f,1]) for an integer f is attached to
Q(ζf ) and isomorphic to (Z/fZ)∗
, whereas bnrinit(Q, f) is attached to its maximal
real subfield Q(ζf + ζ−1
f ) and isomorhic to (Z/fZ)∗
/(±1);
an irreducible integral monic polynomial defining a primitive element for F.
The function bnrcompositum is particularly useful to build compositums in class field theoretic
terms: given two pairs [bnr1, H1] and [bnr2, H2] attached to abelian fields as above, it
returns a pair [bnr, H] attached to their compositum. This is much more efficient than using
polcompositum to obtain a defining polynomial.
Atelier 2022 (13/01/2022) – p. 4/13