4.7. Computing Galois groups over Q.

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We sketch a practical method for computing Galois groups over Q and similar fields. Our

first result generalizes Proposition 4.4.

Proposition 4.19. Let f(X) be a monic separable polynomial in F[X] of degree m with

distinct roots, and suppose that Gf ⊂ Sm has r orbits with m1, . . . ,mr elements respectively

(so that m = m1 +・ ・ ・+mr); then f factors as f = f1 ・ ・ ・ fr with fi irreducible of degree mi.

Proof. Let α1, . . . , αm be the distinct roots of f(X). For S ⊂ {1, 2, . . . ,m}, consider

fS =

_

i∈S(X −αi). This polynomial divides f(X) in Ff [X], and it is fixed under the action

of Gf (and hence has coefficients in F) if and only if S is stable under Gf . Therefore the

irreducible factors are the polynomials fS corresponding to minimal subsets S of {1, . . . ,m}

stable under G, but such sets S are precisely the orbits of G in {1, . . . ,m}.

Now suppose F is finite, with pn elements say, and let E be the splitting field of f. The

Galois group of E over F is generated by the Frobenius automorphism σ : x → xpn. When

we regard σ as a permutation of the roots of f, then its factors in the cycle decomposition

of σ correspond to the distinct orbits of σ. Hence, if the degrees of the distinct irreducible

factors of f are m1,m2, . . . ,mr, then σ has a cycle decompostion of type

m1 + ・ ・ ・ + mr = m.

Lemma 4.20. Let R be a unique factorization domain with field of fractions F, and let f

be a monic polynomial in R[X]. Let P be a prime ideal in R, and let ￣ f be the image of f

in (R/P)[X]. Assume neither f nor ￣ f has a multiple root. Then the roots α1, . . . , αm of f

lie in R, and their reductions ￣αi modulo P are the roots of ￣ f. Moreover G￣ f

⊂ Gf when both

are identified with subgroups of Sym{α1, . . . , αm} = Sym{￣α1, . . . , ￣αm}.

34 J.S. MILNE

Proof. Omitted—see van derWaerden, Modern Algebra, I, §61 (second edition) or Math

676 (Algebraic Number Theory).

On combining these results, we obtain the following theorem.

Theorem 4.21 (Dedekind). Let f(X) ∈ Z[X] be a monic polynomial of degree m, and

let p be a prime such that f mod p has simple roots (equivalently, D(f) is not divisible by

p). Suppose that ￣ f =

_

fi with fi irreducible of degree mi in Fp[X]. Then Gf contains an

element whose cycle decomposition corresponds to the partition:

m = m1 + ・ ・ ・ + mr.

Example 4.22. Consider X5−X−1. Modulo 2, this factors as (X2+X+1)(X3+X2+1),

and modulo 3 it is irreducible. Hence Gf contains (12345) and (ik)(lmn), and hence also

((ik)(lmn))3 = (ik). Therefore Gf = S5.

Lemma 4.23. A transitive subgroup of H ⊂ Sn containing a transposition and an (n−1)-

cycle is equal to Sn.

Proof. Let (123 . . . n − 1) be the (n − 1)-cycle. By virtue of the transitivity, the transposition

can be transformed into (in), some 1 ≤ i ≤ n − 1. Now the (n − 1)-cycle and

its powers will transform this into (1n), (2n), . . . , (n − 1 n), and these elements obviously

generate Sn.

Example 4.24. Select monic polynomials of degree n, f1, f2, f3 with coefficients in Z such

that:

(a) f1 is irreducible modulo 2;

(b) f2 = (degree 1)(irreducible of degree n − 1) mod 3;

(c) f3 = (irreducible of degree 2)(product of 1 or 2 irreducible polys of odd degree) mod 5.

We choose them to have distinct roots. Take

f = −15f1 + 10f2 + 6f3.

Then

(i) Gf is transitive (it contains an n-cycle because f ≡ f1 mod 2);

(ii) Gf contains a cycle of length n − 1 (because f ≡ f2 mod 3);

(iii) Gf contains a transposition (because f ≡ f3 mod 5, and so it contains the product of a

transposition with a commuting element of odd order;o n raising this to an appropriate

odd power, we are left with the transposition). Hence Gf is Sn.

This gives the following strategy for computing Galois groups over Q. Factor f modulo

a sequence of primes p not dividing D(f) to determine the cycle types of the elements in

Gf—a difficult theorem in number theory, the effective Chebotarev density theorem, says

that if a cycle type occurs in Gf , then this will be seen by looking modulo a set of prime

numbers of positive density, and will occur for a prime less than some bound. Now look up

a table of transitive subgroups of Sn with order divisible by n and their cycle type. If this

doesn’t suffice to determine the group, then look at its action on the set of subsets of r roots

for some r.

See, Butler and McKay, The transitive groups of degree up to eleven, Comm. Algebra 11

(1983), 863–911. This lists all transitive subgroups of Sn, n ≤ 11, and gives the cycle types

FIELDS AND GALOIS THEORY 35

of their elements and the orbit lengths of the subgroup acting on the r-sets of roots;w ith

few exceptions, these invariants are sufficient to determine the subgroup up to isomorphism.

Maple can compute Galois groups for polynomials of degree ≤ 7 over Q. To learn the

syntax, type ?galois;. Magma (the replacement for Cayley) probably knows much more,

but my efforts to obtain a manual for it have been unsuccessful.

See also, Soicher and McKay, Computing Galois groups over the rationals, J. Number

Theory, 20 (1985) 273–281.

36 J.S. MILNE

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