Number fields and discriminants (1/2)
We are interested in number fields K = Q[x]/(P) up to isomorphism. Given a monic
irreducible polynomial P ∈ Z[x], the GP function nfinit, determines invariants of K such as
Its signature (r1, r2), r1 + 2r2 = [K : Q] = deg P.
Its absolute discriminant disc(K) ∈ Z, we shall write ∆K := |disc(K)|.
An integral basis, etc.
The integer disc(K) is congruent to 0 or 1 (mod 4), its sign is (−1)r2 ; it is divisible exactly by
the primes that ramify in K.
Theorem. There are finitely many number fields K (up to isomorphism)
satisfying ∆K < B.
The function mapping P ∈ Z[x] to the number field K = Q[x]/(P) is many-to-one; the GP
function polredabs returns a canonical defining polynomial for K. This is the one given in the
LMFDB for instance.
PARI/GP day 2021 (02/06/2021) – p. 2/14