Galois groups (Sage)

Galois groups of polynomials over the rationals, and how the descent found them.

notebook · Sage · research · by Sagebrush · MIT license

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Galois groups

The Galois group of an irreducible polynomial $f \in Q [x]$ permutes its $n$ roots, so, up to the numbering of the roots, it is one of the transitive groups $n$ T $k$ of degree $n = de g f$ . Sagebrush computes it for degrees $1$ to $13$ , $17$ , $19$ and $23$ (PARI stops at $11$ ), entirely in this page:

Frobenius cycle types. For a prime $p$ not dividing the discriminant, the degrees of the factors of $f$ mod $p$ are the cycle lengths of an element of the Galois group (Dedekind). With the parity of the group (is the discriminant a square?) this leaves a few candidate groups, often just one.
Stauduhar's descent. Otherwise, starting from $S_{n}$ or $A_{n}$ : for each maximal subgroup $K$ that is still a candidate, an invariant $F$ of $K$ is evaluated at the roots, computed $p$ -adically, permuted by coset representatives. The group lies in a conjugate of $K$ exactly when a value is a rational integer of known size, and only cosets fixed by the Frobenius at $p$ can work.

The 508 transitive groups of these degrees, and the maximal subgroups of each, were computed by Sagebrush itself; only their standard numbering and names were matched against GAP's library.

R.<x> = QQ[]
for f in [x^2 + 1, x^3 - 2, x^4 - 2, x^5 - 2, x^5 - x - 1, x^6 + x^5 + x^4 + x^3 + x^2 + x + 1]:
    G = f.galois_group()
    print(f, '   ', G.transitive_label(), G.order(), G.name())

x^2 + 1     2T1 2 S2

x^3 - 2     3T2 6 S3
x^4 - 2     4T3 8 D(4)
x^5 - 2     5T3 20 F(5) = 5:4
x^5 - x - 1     5T5 120 S5
x^6 + x^5 + x^4 + x^3 + x^2 + x + 1     6T1 6 C(6) = 6 = 3[x]2

K.<a> = NumberField(x^6 - 2)
K.galois_group(), K.is_galois()

(Galois group 6T3 (D(6) = S(3)[x]2) with order 12 of x^6 - 2, False)

How it was found

The log lists each descent step: the maximal subgroup, its index, how many cosets fixed by the Frobenius gave an integer, and the invariant used (a sum over an orbit of monomials, or a product of differences of roots or of block sums). Repeated values are separated by a Tschirnhausen transformation of the roots.

G = (x^8 - 2).galois_group()
G.galois_log()

discriminant is not a square
5 Frobenius cycle types from 120 primes leave 12 candidates: 8T6, 8T8, 8T15, 8T23, 8T26, 8T27, 8T35, 8T40, 8T43, 8T44, 8T47, 8T50
p-adic roots: p = 7, unramified degree 2, Frobenius cycle type [2, 2, 2, 1, 1]
8T47 in 8T50: repeated values, Tschirnhausen transformation -2x^2 - x
8T50 -> 8T47 (index 35, 1 of 3 Frobenius-fixed cosets integral, invariant of degree 4 with 2 terms, value 64), proven
8T47 -> 8T35 (index 9, 1 of 3 Frobenius-fixed cosets integral, invariant of degree 2 with 4 terms, value 0), proven
8T35 -> 8T26 (index 2, 2 of 2 Frobenius-fixed cosets integral, invariant of degree 6 with 0 terms, value 2048), proven
8T15 in 8T26: repeated values, Tschirnhausen transformation -x^2
8T15 in 8T26: repeated values, Tschirnhausen transformation x^5 - 2x^4 - x^3 + 2x
8T26 -> 8T15 (index 2, 2 of 2 Frobenius-fixed cosets integral, invariant of degree 4 with 8 terms, value 576), proven
8T15 -> 8T8 (index 2, 2 of 2 Frobenius-fixed cosets integral, invariant of degree 3 with 8 terms, value -128), proven
8T8: no maximal subgroup is a candidate

Degrees 12 and 13

Here PARI gives up. The 13th cyclotomic polynomial has the cyclic group $C_{12}$ , $x^{12} - 2$ has a group of order $48 = 12 \cdot φ (12)$ , and compositions $g (h (x))$ have imprimitive groups, whose block systems correspond to the subfields. In prime degree $13$ every transitive group is primitive: $x^{13} - 2$ has the Frobenius group $F_{156} = F_{13} ⋊ F_{13}^{\times}$ .

for f in [sum(x^i for i in range(13)), x^12 - 2, x^12 + 3, (x^4 - 2)(x^3 - 3*x + 1), (x^6 - 2)(x^2 + x), x^12 - x - 1, x^13 - 2]:
    G = f.galois_group()
    print(G.transitive_label(), G.order(), G.name())

12T1 12 C(4)[x]C(3)

12T28 48 D(4)[x]S(3)

12T15 24 1/2[3:2]dD(4)

12T274 10368 [S(3)^4]D(4)=S(3)wrD(4)

12T193 768 [2^6]D(6)=2wrD(6)

12T301 479001600 S12
13T6 156 F_156(13)=13:12

The Mathieu group $M_{23}$

In August 2026, Huang, Jackson, Lee, Poonen, Pries and Zhang proved that $M_{23}$ , the last of the 26 sporadic simple groups not yet known to occur, is a Galois group over $Q$ (arXiv:2608.08538), with this polynomial of degree $23$ . Here the cycle types of Frobenius elements rule out every group but $M_{23}$ and $A_{23}$ (rigorously), and one resolvent, the sum over the $253$ blocks of the Steiner system $S (4, 7, 23)$ preserved by $M_{23}$ of the products of the roots in a block, picks $M_{23}$ : of the $[A_{23} : M_{23}] \approx 1.3 \cdot 1 0^{15}$ cosets only the few fixed by a Frobenius element need to be tried. That last step is a high-confidence computation, not a proof (the paper's proof is geometric).

f = x^23 - 184*x^21 - 1150*x^20 + 26151*x^19 + 18400*x^18 - 1808490*x^17 + 1545462*x^16 + 67672923*x^15 - 42732528*x^14 - 1333395744*x^13 + 290615166*x^12 + 10550424369*x^11 + 3700476348*x^10 + 35123826654*x^9 - 194398310718*x^8 - 1023887308293*x^7 + 3961650395556*x^6 + 1949980486716*x^5 - 28142323927002*x^4 + 53599151839311*x^3 - 46185312415788*x^2 + 19169943578802*x - 3150159884154
G = f.galois_group()
G, G.order(), G.name()

(Transitive group number 5 of degree 23, 10200960, 'M(23)')

G.galois_log()

discriminant is a square
9 Frobenius cycle types from 120 primes leave 2 candidates: 23T5, 23T6
p-adic roots: p = 29, unramified degree 23, Frobenius cycle type [23]
23T6 -> 23T5 (index 1267136462592000, 1 of 1 Frobenius-fixed cosets integral, invariant of degree 7 with 253 terms, value -2281968)
23T5: no maximal subgroup is a candidate

The transitive groups

There are $1, 1, 2, 5, 5, 16, 7, 50, 34, 45, 8, 301$ transitive groups of degrees $1$ to $12$ up to conjugacy (OEIS A002106). Number $295$ of degree $12$ is the Mathieu group $M_{12}$ , which is $5$ -transitive.

[TransitiveGroups(n).cardinality() for n in range(1, 13)]

[1, 1, 2, 5, 5, 16, 7, 50, 34, 45, 8, 301]

M = TransitiveGroup(12, 295)
M, M.name(), M.order(), M.transitivity()

(Transitive group number 295 of degree 12, 'M(12)', 95040, 5)

Proofs

A step is proven when its value is shown to be an integer (by a norm bound, which needs high $p$ -adic precision) and to differ from the values at all other cosets. By default the steps that are cheap to prove are proven. With proof=True, all of them are, however long that takes. Either way, the answers agree with Magma on thousands of test polynomials.

G = (x^12 - 2).galois_group(proof=True)
G, G.proven

(Transitive group number 28 of degree 12, True)

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