Inverse Galois Problem

PCMI 2021 Graduate Summer School Program - Number Theory Informed by Computation - July 26-30, 2021

The Inverse Galois Problem asks whether every finite group occurs as a Galois group over Q. This question and its variant over Q(t) have driven a lot of research, and they are very interesting from a computational perspective as well. There are a variety of methods of constructing Galois groups, and they draw techniques from algebraic geometry, p-adic numbers, the theory of finite groups, Kummer theory, braid groups, group cohomology, and elliptic and modular curves, to name a few. In these lectures I will try to give an overview of the area, and explain where we stand. We will see how the various methods actually work for specific small groups, and get a feel for how far they can be pushed and what are the obstacles to solve the problem in general.

§ Lectures
§ Magma package
§ Exercises
§ Research problems
§ Recommended reading

Lectures

Lecture 1.pdf [video] - Galois groups & transitive groups, Inverse Galois Problem over Q and Q(t), Brief History, Hilbert's Irreducibility Theorem
Lecture 2.pdf [video] - Noether's method, Quotient groups and Extension Problem, Witt's theorems on C₄ and Q₈, Cyclic groups, Split and non-split extensions
Lecture 3.pdf [video] - Split extensions by abelian groups, Examples, Semiabelian groups, SL₂(F₃)
Lecture 4.pdf [video] - Descent to subgroups, Rigidity

Magma package

The package Qt.m provides basic functionality for working with families of Galois groups over Q in Magma. A family is given by a polynomial in x with coefficients in Z[a,b,...], and the function Family takes such a polynomial or a string that defines it and constructs the family. Here is an example of a 1-variable D₄ family:

Attach("Qt.m");                   // Attach package (if it is in the current directory)
F:=Family("x^4+a*x^3+a*x+1");     // D4 family over Q(a)
F;

D4 (1): x^4 + a*x^3 + a*x + 1         ← (1) stands for 'degree 1 in a'

The code computes the Galois group and checks that the family is regular, that is has no non-trivial constant extensions of the ground field (and returns false if not). Also, Family(d,j) returns a pre-defined 1-parameter regular family (the best one I am aware of) for every transitive subgroup G=dTj of S_d with d≤10: Family(7,2); // a family for 7T2 = D7 acting on 7 points D7-7t2 (1): x^7 - x^6 - a*x^5 + 2*3*x^4 + (2*a+39)*x^3 + 17*x^2 - a*x + 2 Basic invariants: G:=Group(F); G; // Galois group

Permutation group acting on a set of cardinality 4
Order = 8 = 2^3
(1, 2)(3, 4)
(2, 4)

GroupName(G); D4

R<a>:=Ring(F);            // Underlying ring Z[a] 
U<x>:=PolynomialRing(F);  // Polynomial ring Z[a][x]
Polynomial(F);            // Polynomial in U that defines F

x^4 + a*x^3 + a*x + 1 Subfields: F:=Family("x^4+a*x+b"); F; S4 (1,1): x^4 + a*x + b Subfield(F,"A4"); // Quadratic subfield cut out by A4 x^2 + 27*a^4 - 256*b^3

H:=Sylow(Group(F),2);   // H = D4 < S4 of index 3
Subfield(F,H);          // Cubic subfield cut out by H

x^3 - 16*b*x - 8*a^2 Changing variables, discriminant, specializations: F<a,b>:=Family("x^6+a*x^2+b"); F; // C2*S4 family C2*S4 (1,1): x^6 + a*x^2 + b Discriminant(F: uptosquares); -b ChangeVariables(F,[a,-b^2]); // Galois group drops to S4 S4 (1,2): x^6 + a*x^2 - b^2 ChangeVariables(F,[a,b^2]); // Not regular as contains Q(i) false Wreath product: C2:=Family(2,1); C2; C2-2t1 (1): x^2 + a C3:=Family(3,1); C3; C3-3t1 (1): x^3 - a*x^2 - 3^2*x + a WreathProduct(C2,C3); // C2wrC3 = C2^3:C3 = C2*A4 C2*A4-C2wrC3 (1): x^6 + a*x^4 - 3^2*x^2 - a WreathProduct(C2,C2: nocheckregular); D4-C2wrC2 (1): x^4 - a ← clearly not regular Families for arbitrary cyclic groups and split extensions by cyclic groups (covered in the lectures): Family("C5");

C5-cyc (10): x^5 - 2*5*(a^4-a^3+a^2-a+1)*x^3 - 5*(4*a^2-2*a-1)*(a^4-a^3+a^2-a+1)*x^2 - 5*(a+1)*
	(3*a^3-6*a^2+4*a-2)*(a^4-a^3+a^2-a+1)*x - (a^4-a^3+a^2-a+1)*(4*a^6-6*a^5-5*a^4+10*a^3-10*a^2+9*a-1)

G:=SymmetricGroup(3);    // Construct an S3-family that contains Q(sqrt a)
N:=Sylow(G,3);           // 1 → C3 → S3 → C2 → 1 is a split extension by a cyclic group C3
Extension(Family("x^2-a"),G,N);

S3-lift (3): x^3 - 3*(a^2+a+1)*x + 2*(a-1)*(a^2+a+1)

Exercises

(Witt's theorem) Let K(√d)/K be a quadratic extension with char K≠2. We will show that it can be embedded in a C₄-extension F/K if and only if d is a sum of two squares in K.

Suppose K(√d) is contained in a C₄-extension F/K, say F=K(√d)(√w) with w in K(√d) and Gal(F/K)=<σ>=C₄. By considering σ(√w)/√w, show that -1 is a norm from K(√d) to K.
Show that -1 is a norm from K(√d) to K if and only if d is a sum of two squares in K.
Suppose d is a sum of two squares in K. By considering the family F:=Family(4,1); F; C4-4t1 (1): x^4 + a*x^3 - 2*3*x^2 - a*x + 1 Discriminant(F: uptosquares); a^2 + 16 show that K(√d) is contained in a C₄-extension of K.

An n-variable G-family F over Q(a,b,...) is said to be generic if every Galois extension L/K in characteristic 0 with Galois group G is its specialisation; that is, there are r₁,...,r_n in K such that L is the splitting field of F(r₁,...,r_n)(x) over K.

Show that (x²-a)(x²-b) is a generic family for G=C₂xC₂.
By considering the differences ±√a±√b and some simplifications, show that x⁴+ax²+b² is also a generic family for G=C₂xC₂, this time given by an irreducible polynomial.

Let G=Dic3=C4:C3 be the dicyclic group of order 12. It is a subdirect product of C₄ and S₃, that is G has normal subgroups N₁ and N₂ with trivial intersection, and G/N₁=C₄, G/N₂=S₃. The N_i do not generate G (else it would be a direct product), but an index two subgroup. By Galois correspondence, a Dic₃-extension is a compositum of a C₄-extension and an S₃-extension whose intersection is quadratic. Use this to construct a regular G-family over Q(a).

Write a script to list all non-semi-abelian groups of order ≤120.

Construct a regular D₄-family over Q(a) in 3 different ways:

As a wreath product
Using that D₄ is a split extension by a cyclic group
Descent from S₄

(One could also use elliptic curves with a 4-torsion point, as for D₅ below.)

C.J.Williamson on p.166 of this paper gives a nice construction of D₉ over Q(a): take an elliptic curve E with a 9-torsion point over Q, and the corresponding 9-isogeny E→E'. Then the extension of function fields Q(E)/Q(E') is cyclic of degree 9, Q(E')/Q(x) is quadratic (and regular), and one can show that Q(E)/Q(x) has Galois group D₉. Use a similar approach to construct a regular family for D₅ over Q(a). (You may find Magma functions T:=SubgroupScheme(E,polynomial in x) and Eprime,phi:=IsogenyFromKernel(T) useful.)

Prove a special case of Hilbert's result on A_n: construct a regular family over Q(a) with Galois group A_n for n=5 and n=6, say. (For instance, consider the family xⁿ+ax+b and analyze its discriminant.)

The group G=C₂.D₈ is a subdirect product of C₄ and D₈ (see Question Q3). Use this to construct a regular G-family over Q(a), and from here construct a regular Q₁₆-family over Q(a). In fact, this approach works for Q_2ⁿ for all n, which is non-trivial - see Jensen-Ledet-Yui's book, p. 141, where this is stated as a research problem.

Consider the following family (obtained as a wreath product) for the group D₄:C₂ of order 16: D4:C2 (2,4): (b^2+4)^2*x^8 - (b^2+4)^2*x^6 - (b^2+4)*(a*b^2+2*a-1)*x^4 + a*(b^2+4)*x^2 + a^2 The group D₄:C₂ has a unique subgroup Q₈ of index 2. Check that the corresponding quadratic extension of Q(a,b) is given by x^2 + (4*a^2*b^2 + 16*a^2 + a*b^2 + 4*a) = 0 This equation defines a surface in P³. Using, for example, Magma's ClassifyRationalSurface and ParametrizeProjectiveSurface functions, find a parametrisation P²→S. Use this to construct a family of Q₈-extensions over Q(a,b,c) whose 3 quadratic subfields are Q(√a²+b²), Q(√a²+b²+c²) and Q(√(a²+b²)(a²+b²+c²)). This family was first constructed by Fujisaki.

Find a rigid Galois stable triple of conjugacy classes in the alternating group A₅.

Research problems

Groups with no known realisations

There are many papers on the realisation of the group SL₂(F₅), a double cover of A₅, but no explicit family over Q(a) is known. See Feit (p232, Problem 1), Mestre, Crespo and Schneps. Is it possible to construct an SL₂(F₅)-family over Q(a)?

The are two transitive groups on ≤23 points for which the Inverse Galois Problem is not known over Q, namely 17t7=SL₂(F₁₆).C₂ and the sporadic simple group M₂₃; see Kluners-Malle database - missing groups. Is it possible to realise them?

Degree 1 families

Hilbert's and Serre's constructions of A_n fields give degree 2 families over Q(a). However, for n=3,5,6,7,9,10 there are degree 1 families: for n:=3 to 10 do Family(n,NumberOfTransitiveGroups(n)-1); end for;

C3-3t1 (1): x^3 - a*x^2 - 3^2*x + a
A4-4t4 (2): 3*x^4 + 2^2*x^3 + 3*a^2+1
A5-5t4 (1): x^5 + 3*5^2*x^3 + a*x^2 + 3*a
A6-6t15 (1): x^6 + 3*x^5 + 3*5*x^4 + a*x^3 + 2^2*3*x - 2^2
A7-7t6 (1): x^7 - 2*x^6 + (a+9)*x^4 + a*x^3 + 2*x - 1
A8-8t49 (2): 7*x^8 + 2^3*x^7 + 7*a^2+1
A9-9t33 (1): x^9 - 2^2*3^2*x^5 + 2^2*3^2*5^2*x + a
A10-10t44 (1): x^10 + 3*x^5 + a*x - 2^2

Is there an explanation for these, and do they generalise to other n? Do they exist for infinitely many n? Do degree 1 families exist for A₄ and A₈?

This is the same question for small soluble groups: are there degree one families for C₂³, C₂xC₄ or SL₂(F₃), and, if not, why not?

Even for cyclic groups C_p of prime order p the best degree is unclear. I will explain in the course how to construct a family for C_p of degree p(p-1)/2-1 in a, but for p=3,5,7 this is not best possible: [Family(p,1): p in [3,5,7]];

[
C3-3t1 (1): x^3 - a*x^2 - 3^2*x + a,
C5-5t1 (4): x^5 - a^2*x^4 - 2*5*(2*a^2+5)*x^3 + 2*a^2*(2*a^2+5)*x^2 + (24*a^4+300*a^2+625)*x + a^2*(36*a^2+95),
C7-7t1 (10): x^7 + (a^3+2*a^2-a+13)*x^6 + 3*(a^2-a+2)*(a^3+a+9)*x^5 + (3*a^7-9*a^6+27*a^5-22*a^4+6*a^3+84*a^2-121*a+75)*x^4 
  + (a^2-a+2)*(a^7-5*a^6+15*a^5-32*a^4+20*a^3+14*a^2-113*a-1)*x^3 + (-a^10+5*a^9-25*a^8+61*a^7-126*a^6+117*a^5-58*a^4-155*a^3+
  168*a^2-80*a-44)*x^2 + (-a^10+8*a^9-30*a^8+75*a^7-102*a^6+89*a^5+34*a^4-56*a^3+113*a^2+42*a-17)*x + 
  a^9-7*a^8+23*a^7-42*a^6+28*a^5+19*a^4-60*a^3-2*a^2+16*a-1
]

Can one improve the upper bound or to prove a lower bound for a possible degree?

Torsion points on algebraic groups

The Galois group of x^p-a for p prime is the Frobenius group F_p, making F_p one of the easiest Galois groups to realise over Q. However, this obvious family over Q(a) is not regular, because the splitting field contains pth roots of unity, and it appears to be surprisingly hard to construct regular F_p families of small degree. In fact, of the 651 transitive groups of degree ≤15 the worst family I have is for F₁₁=11t4, of degree 660 in a. Nevertheless, for some p there are families such as F7 (2): a*x^6*(a*x-1) + (7*a^2+13*a+7)*(x^5+x^4+x^3+x^2+x+1 + a) that are regular and quite symmetric, suggesting that there could be a construction out there for all (or at least infinitely many?) primes p. Is that the case? Specifically, is there a better family for F₁₁?

The situation with GL₂(F₃) is similar. That there are numerous GL₂(F₃)-extensions on LMFDB with small coefficients (cf. e.g. Q₈), suggesting that there are good families. Indeed, if E/Q an elliptic curve with c-invariants c₄, c₆, then the 3-torsion field Q(E[3]) is the splitting field of x⁸-6c₄x⁴-8c₆x²-3c₄². As a very special case of Serre's open image theorem, this is a GL₂(F₃)-family over Q(c₄,c₆). It is great, but not regular, again because of roots of unity. By the properties of the Weil pairing, Q(E[3]) contains Q(ζ₃) as a quadratic subfield cut out by SL₂(F₃): F:=Family("x^8-6*a*x^4-8*b*x^2-3*a^2": nocheckregular); F; GL(2,3) (2,1): x^8-6*a*x^4-8*b*x^2-3*a^2 Subfield(F,SL(2,3)); x^2 + 3 What is the way to construct good regular GL₂(F₃)-families, or, even better, regular GL₂(Z/nZ)-families for all n?

Algorithms

Stauduhar's algorithm, especially as descrbed and implemented by Fieker and Klüners is an amazingly efficient algorithm to compute Galois groups of polynomials over Q of arbitrary degree. Is it possible to adapt it so that it works for families, with provably correct results?

If G=A:H is a split extension with A abelian, every H-extension (say, over Q(a)) can be lifted to a G-extension. I will explain an explicit version of this result in the lectures for A cyclic, and it is fundamental for constructing families of soluble groups. Is it possible to implement an efficient analogue for when A is non-cyclic? One interesting example is the group G=C₃².He₃ =(C₃xC₉):C₉; here A=C₃xC₉ and H=C₉. If it can be constructed, then so can be its quotient C₃.He₃ of order 81, which is the only group of order 64<n<108 for which I do not know a family over Q(a).

If H<G is a subgroup, then descending a G-family to an H-family is a question of finding rational curve (or a higher-dimensional rational variety) on the hypersurface that defines the corresponding subfield. Is there a general method to find rational curves on a hypersurface?

For some Galois groups it is easy to construct families over Q(a) by interpolating examples over Q. For example, take f₁, f₂ from the list of the first 10 polynomials with Galois group PSL₂(F₇) from LMFDB and constant term 1. The degree 1 one parameter family f₁+a(f₂-f₁) through f₁ and f₂ turns out to have the group PSL₂(F₇) over Q(a) several times:

R<x>:=PolynomialRing(Rationals());
L:=[   
  x^7-x^6-3*x^5+x^4+4*x^3-x^2-x+1,       x^7-2*x^6-x^5+4*x^4-3*x^2-x+1,
  x^7-x^6-4*x^5+x^4+4*x^3-3*x+1,         x^7-x^6+2*x^5-3*x^4-4*x^3+2*x^2+3*x+1,
  x^7-x^6+2*x^5+2*x^4-5*x^3+7*x^2-5*x+1, x^7-3*x^6+3*x^5+3*x^4-13*x^3+17*x^2-10*x+1,
  x^7-2*x^5-4*x^4-x^3+5*x^2+5*x+1,       x^7-x^6+x^5-4*x^4+4*x^3-5*x^2+2*x+1,
  x^7-x^6-3*x^5-x^4+2*x^3+4*x^2+4*x+1,   x^7-2*x^6+4*x^4-3*x^3-4*x^2+2*x+1
];
for f1,f2 in L do
  F:=Family(Sprintf("%o+a*(%o)",f1,f2-f1));
  if F cmpne false and IsIsomorphic(Group(F),PSL(2,7)) then F; end if;
end for;

PSL(2,7) (1): x^7 + (-a-1)*x^6 + 3*(a-1)*x^5 + (3*a+1)*x^4 + (-7*a+4)*x^3 + (-3*a-1)*x^2 + (3*a-1)*x + 1
PSL(2,7) (1): x^7 - x^6 + (5*a-4)*x^5 + (-5*a+1)*x^4 + 2^2*x^3 - 5*a*x^2 + (5*a-3)*x + 1
PSL(2,7) (1): x^7 - a*x^6 + (-a-2)*x^5 + (3*a-4)*x^4 + (3*a-1)*x^3 + (-a+5)*x^2 + (-a+5)*x + 1
PSL(2,7) (1): x^7 - x^6 + (-5*a+1)*x^5 + (5*a-4)*x^4 + 2^2*x^3 + 5*(a-1)*x^2 + (-5*a+2)*x + 1
PSL(2,7) (1): x^7 + (a-1)*x^6 + (a-3)*x^5 + (-3*a-1)*x^4 + (-3*a+2)*x^3 + (a+4)*x^2 + (a+4)*x + 1
PSL(2,7) (1): x^7 + (a-2)*x^6 - 3*a*x^5 + (-3*a+4)*x^4 + (7*a-3)*x^3 + (3*a-4)*x^2 + (-3*a+2)*x + 1

Such families are often the best known families for a given group as they have small coefficients, and they give an excellent sanity check for the more theoretical constructions for how good they are. For PSL₂(F₇), this can be pushed much further - see Malle's paper, §3.1, §4. But for other groups, for instance GL₂(F₃), it seems quite difficult to find even one family from thousands of examples. Is there a good algorithm to detect or construct 1- (or more) parameter families from a list of examples?