Galois theory (2/3)
Some GP functions:
polgalois (with galdata package) returns the isomorphism type of Gal(P) for
deg P 6 11; we advise to set new_galois_format to 1.
nfsplitting returns a defining polynomial P̂ for the splitting field of P. It runs in
polynomial time in the degree of P̂. Of course, if deg P = n then deg P̂ may be as large
as n! but this works well if K̂ is not too large, say a few seconds for n̂ := [K̂ : Q] < 1000;
a multiplicative upper bound for n̂ helps a lot.
for a Galois number field K/Q, galoisinit returns the Galois group of K as a structure
allowing basic Galois theory: conjugacy classes, character table, subgroups and fixed fields.
(The group must be “weakly super solvable”, i.e., have a normal series H0 = {1} ⊳ H1 · · · ⊳ Hm with cyclic
factors Hi+1/Hi, such that
G/Hm ≃ {1} , A4, S4, or 9T9 = (C3 × C3) ⋊ C4 .
Most small groups have this property.)
shortcut: galoissplittinginit ≈ nfsplitting + galoisinit.
PARI/GP day 2021 (02/06/2021) – p. 5/14