Galois theory (1/3)
Let K = Q[x]/(P) be a number field of degree n and let K̂ be the splitting field of P: K̂/Q is
the Galois closure of K/Q. We write G = Gal(K̂/Q) = Gal(P) for its Galois group. We can
view G as a transitive subgroup of Sn (acting on the roots of P). Transitive subgroups of Sn up to
conjugacy are classified for small n, including all n 6 47:
n = 2. We have 1 group (C2)
n = 3. We have 2 groups (C3 = A3, S3)
n = 4. We have 5 groups (C4, C2 × C2, D4, A4, S4)
n = 5. We have 5 groups (C5, D5, F5 = C5 ⋊ C4, A5, S5)
n = 8/16/32. We have 50 / 1,954 / 2,801,324 groups.
We write nTk for the k-th transitive subgroup of Sn in this classification. For fixed n, the group
order increases with k; in particular, the last two elements in the series are An and Sn (and Cn
comes first for n 6= 32).
PARI/GP day 2021 (02/06/2021) – p. 4/14