Galois theory (3/3)
Conjecture (Inverse Galois problem). Every transitive permutation group G
occurs as a Galois group over Q.
Conjecture (Malle). Let n > 2. For every transitive G ֒→ Sn, there exist
computable integers a(G) > 0, b(G) > 0 and some positive constant c(G) > 0
such that
#
n
K/ ≃, Gal(K̂/Q) = G, ∆K < B
o
∼ c · B1/a
(log B)b
.
For many small groups G, the GP function nflist returns (defining polynomials for) number
fields K/Q with given Galois group Gal(K̂/Q) = G, possibly fixing signatures and/or some
resolvent subfield of K̂. In good cases, all fields with ∆K ∈ [A, B].
It relies on the optional package nflistdata being installed.
PARI/GP day 2021 (02/06/2021) – p. 6/14