The Regular Inverse Galois Problem

The R(egular) I(nverse) G(alois) P(roblem)

This exposition relates four famous problems, denoted below by I(nverse) G(alois) P(roblem), the R(egular) I(nverse) G(alois) P(roblem), the R(egular) S(plit) E(mbedding) P(roblem) and the Beckmann-Black Problem. The RSEP Conjecture implies all of the first three, which are known for many non-trivial subfields of the algebraic numbers. Running over all finite groups, none is known for any number field. The most successful variant on the RIGP is the Nielsen-RIGP. We explain it, then give examples of its successes and its cutting edge problems.

Table of Contents:
I. INTRODUCTION AND THE IGP:
I.1. INTRODUCTION TO THE MONODROMY METHOD:
I.2. THE IGP:
II. THE RIGP AND THE R(EGULAR) S(PLIT) E(MBEDDING) P(ROBLEM):
II.1. THE RIGP AND ITS USE:
II.2. R(EGULAR) S(PLIT) E(MBEDDING) P(ROBLEM):
II.3. OBSERVATIONS ON THE RSEP:
II.4. THE BECKMANN-BLACK CONJECTURE:
III. THE NIELSEN-RIGP:
III.1. CONJUGACY CLASSES GIVE A REFINED CONJECTURE:
III.2. CONJOINING CLASSES:
III.3. WHEN YOU CAN'T CONJOIN CLASSES:
III.4. DIHEDRAL GROUP PROPERTIES GENERALIZE:
IV. ALTERNATING GROUP ILLUSTRATIONS:
IV.1. COHERENT COLLECTIONS OF SPLIT ABELIAN A_n EXTENSIONS:
IV.2. COHERENT COLLECTIONS OF p-FRATTINI ABELIAN A_n EXTENSIONS:

I. INTRODUCTION AND THE IGP:

I.1. INTRODUCTION TO THE MONODROMY METHOD: Thanks to the Braid group monodromy method (say, as in inv_gal.html), many families of simple groups have regular realizations, and therefore, so do abelian semidirect products with these groups (as in §IV). The groups, however, that generate the biggest mysteries, are the p-Frattini covers of p-perfect groups, the standout item of the list of unsolved problems . This is the elementary formulation of the appearance, and properties, of Modular Towers, the construction that generalizes the theory of modular curves. Modular Towers plays on the discussion below – on the Nielsen-RIGP – where dihedral groups point the way to the depth of the RIGP. This is despite how first year graduate algebra easily constructs regular realizations for them (compare §III.2 and §III.3).

The full scope of the RIGP entertains any problem that requires understanding the coefficients of algebraic equations. An example success appears in the solution of Davenport's Problem – describing polynomial pairs whose ranges over the residue class fields of a number field are the same for almost all primes. Its solution covered great territory in relating to general techniques for solving algebraic equations and to the formulation of the Genus 0 Problem. We have prepared UMStory.html and other files on the published implications of Davenport's Problem. You could call that "what we learned from polynomials in one variable." In this page, however, we stay with just the RIGP. As with dihedral groups, refined questions show new lessons evolve from what seemed long known territory.

I.2. THE IGP: Assume F is a field extension of the rational numbers Q of finite degree: as a vector space over Q its dimension [F:Q] < ∞. Let Q^– be the set of all algebraic numbers: complex numbers that are zeros of some polynomial f ∈ Q[x]. There are exactly [F:Q] distinct maps (embeddings) of F into Q^– that preserve the multiplication and addition structure on F.

We say F/Q is Galois if all these embeddings end up back in F. More generally, call the minimal field, F^{^}, containing all the images of the embeddings of F the Galois closure of F/Q. When F/Q is Galois, the embeddings form a group under composition. That group, G(F/Q), is the (Galois) group of F. This idea of Galois over a field works with any starting field L replacing Q. It too has a minimal field containing elements that are zeros of any positive degree polynomial with coefficients in L, an algebraic closure L^– of L.

One version of the I(nverse) G(alois) P(roblem) asks: Is each finite group G the (Galois) group of some F/Q?

A stronger version asks for each G: For each finite extension F/Q, is there an L/F with G=G(L/F)?

The profinite group of all automorphisms of F^– fixed on F is the absolute group G_F. A group is profinite if it is the projective limit of finite groups. Absolute Galois groups are the main example of these.

Here is another formulation of the IGP for G: For each finite extension F/Q, G is a quotient of G_F.

II. THE RIGP AND THE R(EGULAR) S(PLIT) E(MBEDDING) P(ROBLEM):

II.1. THE RIGP AND ITS USE: The R(egular) IGP is similar, except it changes Q to Q(z), rational functions in an indeterminate z. Further, it asks for each (finite group) G: Is there a field F regular over Q(z) (F∩Q^–=Q) with G=G(F/Q(z)). The advantage of the RIGP: Once done for G over Q(z), applying H(ilbert's) I(rreducibility) T(heorem) gives the stronger form of the IGP for G. In this exposition we consider primarily the RIGP over Q. So long as we restrict to characteristic 0, we need no new ideas – though perhaps much notation – to extend that consideration to any characteristic 0 field.

Galois Theory is the translater par excellence of problems stated in algebraic equations to problems stated in group theory. While Group Theory isn't easy, it is a well-oiled machine with many effective classification results. Equations are usually impossible to solve: Their solutions don't have expressions in functions with known properties. Studying the solutions of each new equation from the beginning seems hopeless. Yet, some equations do have expressions in functions that repeatedly arise.

In practice, researchers – handling present-day problems – who must solve equations stick near previously understood equations. Therefore they combine Galois theory with methods for identifying equation solutions with classical functions. The papers of Articles: Finite fields, Exceptional Cover and Motivic Poincaré series have many of that type, with particular exemplars (like "Galois groups and Complex Multiplication" which solves the Schur Conjecture for rational functions by using modular functions) listed in nonabel-cryptology.html.

In addition to applying solutions to the RIGP, researchers consider four other aspects of the RIGP:

Interpretation of it as a problem of finding Q solutions of well-defined equations.
Its relation to Hilbert's Irreducibility Theorem.
Methods for solving it in special cases, starting with the case when G is simple.
Interpreting its relations to other mathematics problems, especially those related to classical equations like modular curves.

We concentrate here on items #1 and #2. Item #3 is the subject of two books: [MM]; and [Vo]. Item #4 is the featured topic in Modular Towers.

II.2. R(EGULAR) S(PLIT) E(MBEDDING) P(ROBLEM):

This conjecture from [DeDes] plays a central role in Inverse Galois Theory:

Conjecture (Debes-Deschamps): Let K be a field. Also let E/K(z) be Galois (not necessarily regular) with (finite group) G and let f: H→ G be an onto homomorphism that splits (there is a section G→ H). Then there is a Galois field extension F/K(z) with group H with these two properties:

the fixed field of ker(f) in F is E; and
F ∩ K^– = E ∩ K^–

#5 defines a proper solution to the embedding problem (E/K(z); f) over K(z). #6 is the condition for a regular solution. So, an alternate formulation of Debes-Deschamps, using this embedding problem language for (G,H), asks this. Does every split finite embedding problem for G_K(z)→ G(E/K(z)) = G have a proper regular solution mapping to H.

II.3. OBSERVATIONS ON THE RSEP: This makes no assumption on the base field K. From this it unifies two major problems from Inverse Galois Theory: The RIGP and Shafarevich's Conjecture. The latter says the absolute Galois group of the field Q^ab – obtained by adjoining all complex numbers for which some power is 1 (the cyclotomic numbers) – is profree.

We say a field K is projective if G_K is a projective (profinite) group (see [FrJ, Chap. 22] or the definition file p-Frattini-cov.html). The RSEP implies a more general conjecture of [FrVo]:

Conjecture (Fried-Völklein): If K is a Hilbertian countable field with G_K projective, then G_K is profree.

A P(seudo)A(lgebraically)C(losed) field is automatically projective [FrJ, Thm. 11.6.2] (originally due to Ax; the name is due to Frey). So, the Main result of [FrVo], that the conclusion of this conjecture holds for PAC Hilbertian fields, is a special case.

The argument that RSEP implies Fried-Völklein has three steps [DeDes]:

As G_K is projective, an Iwasawa result says we need only show each split finite embedding problem for G_Khas a proper solution.
From the RSEP Conjecture, such an embedding problem has a solution over K(z).
Then, Hilbertianity of K allows specializing z in K for a solution of the original embedding problem over K.

The Shafarevich conjecture follows as Q^ab is classically known to satisfy the assumptions of the Fried-Völklein conjecture (see exposition [FrJ, Prop. 11.6.6 and Thm. 13.9.1]). The RSEP Conjecture was proved by Pop [Po] for K a large field: Every smooth absolutely irreducible curve over K has infinitely many K points provided it has one. Another topic deserves mention here. An example of a large field, that isn't P(seudo)A(lgebraically)C(losed) (as appears in [FrVo]) is the field of totally real algebraic numbers, a classical field whose absolute Galois group we know: It is countably freely generated by involutions (QTotallyReal.pdf).

II.4. BECKMANN-BLACK CONJECTURE: This problem has the virtue of raising the relation between the RIGP and IGP by considering the possibility that regular realizations may be as plentiful as ordinary realizations. Still, there is no known technique – in contrast to the Nielsen-RIGP – for handling this problem over a number field.

Conjecture (Beckmann-Black): Given a finite Galois extension E/K with group G, there exists a regular Galois extension F/K(z), with group G, specializing to E/K at some unramified point z₀.

III. THE NIELSEN-RIGP: Any regular realization G=G(F/Q(z)) of a group G has an attached set of conjugacy classes C={C₁,…,C_r}, each corresponding to one of the branch points of a realizing extension F/Q(z). These conjugacy classes are natural parameters for all reasonable approaches to the RIGP. That makes the Nielsen-RIGP conjecture more precise, right on top of actual work on the RIGP. Its connection to Modular Towers is another advantage of it.

III.1. CONJUGACY CLASSES GIVE A REFINED CONJECTURE: There are two necessary conditions for C to be attached to a Q regular realization of G.

Condition 1: A special case of the B(ranch) C(ycle) L(emma) says C must form a rational union of conjugacy classes: For p any prime not dividing the orders of any elements in C, the set of those elements is closed under putting its members to the p power.

Condition 2: There must be elements g_i∈C_i that generate G, and also satisfy the product-one condition: g₁ ^… g_r = 1. That is, the Nielsen class of (G,C) must be nonempty. We refer to the r-tuple (g₁, …, g_r) as branch cycles in the Nielsen class.

By adding "powers" of conjugacy classes to any classes C, there is always a minimal rational union of classes C^rat containing C: The rationalization of C. The full BCL handles much more, including how you would change Q realizations to consider regular realizations over any characteristic 0 field. Indeed, it has been more practically applied outside the RIGP to identifying the (minimal) fields of definition of covers that aren't Galois.

Now consider any collection of distinct conjugacy classes C'={C'₁,…,C'_u} satisfying these two conditions:

their collection of elements generate G; and
they are themselves a rational union of conjugacy classes.

Conjecture (Nielsen-RIGP): For infinitely many sets of conjugacy classes C supported just in the classes of C', there is a regular realization of G in the Nielsen class of (G,C).

Note: #7 and #8 are necessary for the conclusion of the Nielsen-RIGP.

Consider a (finite or profinite) group G, with order divisible by a prime p. Then, G is called p-perfect if Z/p is not a quotient of G. A perfect group is p-perfect for each prime dividing its order. Simple groups are perfect, but so is any Frattini cover of a perfect group. For a given G, denote by NP_G the collection of primes for which G is not perfect.

Here are some examples. With G=S_n, the symmetric group of degree n ≥ 2, NP_G = {2}. It is also {2} for D_p^k, the dihedral group of order 2p^k, where p is an odd prime. A group is nilpotent if it is a product of its p-Sylows. The primes dividing the orders of a nilpotent group G are exactly those in NP_G. More generally, NP_G consists of the primes dividing the order of the maximal nilpotent quotient U of G.

The exponent of a p group is the maximal power of p for the order of any element in it. If P is a p group, then it's abelianization has the same rank (minimal number of generators) and exponent. From that conclude the following.

Nilpotent Principle: For #7 (generation) to hold, for each p group in U, there is at least one class in C' with elements of order divisible by the exponent of that p group. For #8 (rational union) to hold, there must be at least φ(p^k) – cardinality of the invertible integers mod p^k – such classes.

III.2. CONJOINING CONJUGACY CLASSES: We review some advantages (with bottlenecks) to the Nielsen-RIGP. The distinction with the RIGP is that we are trying to find the role of the conjugacy classes in a regular (really, any) realization. To see why the Nielsen-RIGP is a helpful guide, keep in mind that no one knows all finite groups, not even all p-groups, in any meaningful way. Yet, most mathematicians know all abelian groups, and some even know all simple groups. Note, too, how quickly we come to problems from the literature that, at first, seem unrelated to the RIGP.

Consider a short exact sequence of groups: 1 →Ker →G→G^*→1 and a collection C' of the distinct conjugacy classes of G. Here is one of the simplest observations in the game:

Quotient Principle: A regular realization of G automatically gives a regular realization for its quotient G^*.

That is, if L^{^}/Q(z) is a regular Galois extension with group G, then the fixed field of Ker is a regular Galois extension of Q(z) with group G^*.

Suppose C' is the set of distinct conjugacy classes appearing in this regular realization, so their image, C'_G^* – stripped of any repetitions – in G^* generates it. Further, some conjugacy classes of C' could be in Ker. Denote these C'_Ker, noting the possibility some classes might break into a nontrivial union of classes in Ker. Now consider the possibility of reversing this, where we don't start with C', but rather just with C'_G^* (in G^*) and C'_Ker (in Ker).

Conjoining C'_G^* and C'_Ker Questions: Assume #7 and #8 hold for C'_G^* (in G^*) and for C'_Ker (in Ker).
CQ_a: When is there a natural C' attached by which we can produce solutions to the Nielsen-RIGP for (G,C')?
CQ_b: When is this the right question?

Split example: An excellent example of CQ_a appears in [FrJ, Lem. 16.4.4]: Assume Ker is abelian, and the morphism G→G^* splits. If C'_Ker satisfies #7 and #8, [FrJ, Lem. 16.4.3] produces a regular realization of Ker supporting the conclusion of the Nielsen-RIGP for Ker. Then, if the conclusion of the Nielsen-RIGP holds for (G^*,C'_G^*) you can conjoin those regular realizations to produce C' to give the conclusion of the Nielsen-RIGP for G.

RIGP-splitab.html reviews this. It goes through the regular realization of the general wreath product of groups G^* and M (§ II, RIGP-splitab.html) given the regular realization of G^* and M. There are several subtleties, including one cohomological (the field-crossing argument of Chebotarev). The dihedral group case is when G^*= Z/2 and Ker = Z/p^k.

RIGP-nilp.html lists limitations on this method. It starts by noting that nilpotent regular realizations are cohomologically too intricate to replicate the elementary result above. §III.3 discusses the problems under the Nielsen-RIGP that connect the RIGP and modular curves in the Modular Towers program. It's about dihedral groups as the easiest example of the Frattini p group extensions of any p-perfect group.

III.3. WHEN YOU CAN'T CONJOIN CONJUGACY CLASSES: Now assume in the "Conjoining Questions" above that C'_Ker is empty but C'_G^* satisfies #7 and #8. In the following two cases you can produce a compatible C' (giving generation, #7) by rationalizing (§III) any lifts of C'_G^* to G^*.

Split case: Assume in the Split Example above, Ker is generated by commutators of the form aga^-1g^-1, a∈Ker and g∈C'_G^*.
Frattini case: Assume G→G^* is a Frattini cover; the exact opposite of split: no proper subgroup of G maps surjectively to G^*.

Our simplest example of #9 is G = D_p^k, G^*= Z/2 (p odd; write Z/2 multiplicatively as {±1}) and C'_G^* is the conjugacy class of -1. Our simplest example of #10 is the same G with k≥2, and G^*=D_p.

Two (order two) lifts of -1 to D_p^k will generate D_p^k, if they differ mod p. A regular realization for this case of the Nielsen-RIGP is called an involution realization of D_p^k. The following points are established in [DeFr, §5.2] and put in the context of [Se] in [Fr2, §7]. Fix p, any odd prime.

The BCL (Condition 1 of §III.1) says that the Q regular realizations of D_p^k in §III.2 require an increasingly large number of branch points as k increases. If there is to be any uniform bound with k on the number of branch points, then almost all must be involution relations.
Any involution realization of D_p^k corresponds to a μ(p^k) torsion point on some hyperelliptic Jacobian.
Should there be some bound B on the number of branch points for Q involution realizations of the set {D_p^k}_k=1^∞, then there is a fixed genus g ≤(2B-2)/2, so that for each k, some hyperelliptic Jacobian genus g has a μ(p^k) torsion point.

A μ(p^k) torsion point is one that generates a torsion group of order p^k on which the absolute Galois group of Q acts as if the group is a copy of the p^kth roots of 1. Producing such points is an analog of producing p^kth-power torsion points on a hyperelliptic Jacobian over Q.

A compendium of problems, with references, about p^k torsion points is in [P]. For the case with r=4 branch points for involution realizations there is a well understood relation between p^k torsion points and μ(p^k) torsion points over Q, the territory of Mazur's Theorem limiting such torsion on elliptic curves [DeFr, Thm. 5.1]. For larger (must be even) values of r, the relation is trickier. We engaged an author of papers – F. Leprevost – on this issue in [Fr1, §I.D]. This showed by explicit examples the nuance in the μ(p^k) problem.

III.4. DIHEDRAL GROUP PROPERTIES GENERALIZE: Though not obvious, all observations and conjectures on dihedral groups D_p in §III.3 naturally generalize (combining #9 and #10) to consider any p-perfect finite group and its infinite collection of Frattini extensions with p group kernel (see How MTs arise, in mt-overview.html). Frattini extensions automatically have nilpotent kernel [FrJ, Lem. 22.1.12]. All appear in the theory.

The main conjectures and results interpret as properties of systems of spaces that are modular curve generalizations. Thus, this is a gateway to how much of the RIGP is encoded into the Modular Towers project. Also, why it has so many connections to the S(trong) T(orsion) C(onjecture) (from the theory of Abelian varieties), a topic featured in the Modular Towers TimeLine Modular Towers.

In particular, the STC implies the first diophantine generalization to Modular Towers of a modular curve property: High tower levels have no Q points [CaDe]. The most general result so far has shown this when r=4 (spaces of 4 branch point covers) [CaTa]. §IV compares the two methods toward this general conjecture by their application to extensions of alternating groups. Since little has been proved on the STC, the explicit nature of Modular Towers is leading the way to results.

The only exclusion from this method is when the kernel has order divisible by the primes of NP_G. That is, the approach separates p groups and p-perfect groups. RIGP-nilp.html supports that the analog of Shafarevich's Theorem, the RIGP for p groups, is truly a separate problem, whose adherents use different tools, depend on explicit equations, and have very different motivations.

IV. ALTERNATING GROUP ILLUSTRATIONS:

We list right-at-hand challenges for the RIGP for one example from twoorbit.pdf. See §I of RIGP-splitab.html for the notation for semi-direct product.

A_n Extensions: G^*=A_n, C=C_3^n-1, n-1 repetitions of 3-cycles. For each prime p, we consider various p group extensions, G, of G*. We took 3-cycles rather than other classes, as you will see, because that excludes only the prime 3 from partaking in the major issues.

IV.1. COHERENT COLLECTION OF SPLIT ABELIAN A_n EXTENSIONS: Each alternating group A_n, in its embedding in S_n, acts as permutation matrices on Zⁿ. Sum the elements of the natural basis to get a 1 dimensional submodule invariant under S_n. Quotient by this for an n-1 dimensional Z[A_n] module V_n. For each prime p, consider the quotient V_n/p^kV_n=V_n,p^k.

There are many ways to regularly realize A_n (over Q). One, especially famous, goes back to the 19th century, in Hilbert's paper [Hi] proving his irreducibility theorem. Regular realizations using n-1 3-cycles are the topic of [Me] for n odd. They all derive from a single connected Hurwitz space component with a dense set of Q points in [Fr3, Cor. 5.1]. [Se, §9.3] supplements this for n even, though his Nielsen class is not for (A_n,C_3^n-1), but for one derived from that Nielsen class with n+1 replacing n.

RIGP-nilp.html says more on that choice, though you don't need that here. We now consider several extensions of A_n with abelian p groups as kernels. So long as p≠3, there is a unique conjugacy close in each of these extensions that lifts the class of 3-cycles in A_n. So, for each of these extensions, we unambiguously use the same notation C_3^n-1 as in A_n.

§III of RIGP-splitab.html says each V_n,p^k×^sA_n has a regular realization (over Q). Yet, as in §III.4 – akin to the dihedral case – this realization requires conjugacy classes satisfying #7 and #8 (in V_n,p^k); these classes all have p-power order.

Fix p: When can you expect to use the same regular A_n realization to realize V_n,p^k×^sA_n, using just elements in C_3^n-1 for branch cycles?
Same question as #14, but we don't fix the regular realization A_n from the Nielsen class.

In #14, attempting to restrict to C_3^n-1 in V_n,p^k×^sA_n requires showing condition #7 holds, though #8 is automatic. Although we have a complete new Nielsen class for the extension of A_n, §IV of RIGP-splitab.html says we can satisfy #7 so long as p≠ 2 (and we've already excluded 3). Still, for p fixed, realizing V_n,p^k×^sA_n for all k fails for a conceptual reason: There are no projective sequences of points on a Modular Tower [BaFr, Thm. 6.1]. Indeed, #14, and anything similar to it, for any group G has the proven answer "never."

Recall, however, [Me] provides infinitely many (Q) regular realizations in the Nielsen class for (A_n,C_3^n-1). So, maybe, for fixed p, we might vary the A_n realization in the given Nielsen class for each k to regularly realize V_n,p^k×^sA_n. It is, however, a case of the Main Modular Tower Conjecture that this is impossible. [CaTa, Cor. 1.2] has proven that main conjecture when r=4, so for n=5. A more explicit proof of this case follows from [BaFr, §9.1]. Neither method, yet, can give the expected answer "Never!" for any higher value of n.

IV.2. COHERENT COLLECTIONS OF p-FRATTINI ABELIAN A_n EXTENSIONS: For any prime p ≤ n (but no others), there is a u_n,p (≥ 2), with A_n acting on (Z_p)^u_n,p=U_n,p: denote U_n,p/p^kU_n,p by U_n,p^k. Further, U_n,p^k is the kernel for the maximal p-Frattini extension, A_n,p^k of A_n with abelian kernel of exponent p^k (see p-Frattini covers.html).

Can you imitate the split-abelian result of §IV.1 for A_n,p^k, for all k, allowing conjugacy classes where p divides their orders?
Can you do the previous question in the style of #14 or #15, for p≠3, using only branch cycles in C_3^n-1.

Question #16 is natural, though no one has yet seen how to use the p classes in the kernel of an extension when the extension is not split, say for Frattini covers. Indeed, if you stay with G=A_n, how would you get all its p-Frattini covers, without consideration for what Nielsen classes you use for regular realization of A_n. Avoiding using conjugacy classes with orders divisible by p is for alternating groups the analog of involution realizations for dihedral groups.

For example, the 2-Frattini (resp. 5-Frattini) cover of A₅ is an extension of degree 2⁵ (resp. 5⁶). That has never been regularly realized. The point, however, of using the 3-cycle conjugacy classes in [Me] is they do provide regular realization – without using 2-classes – of the famous degree 2 central extension, the spin cover, of A_n.

On the other hand, consider the answer to question #17 in imitation of the answer for #14 (resp. #15). In the former case, "Never!" and in the latter, as given by the Main Conjecture of Modular Towers, the expected answer is "Never!," though it is only known for these examples when n=5, by both methods mentioned in §IV.1.

Notice, also, that not even for the Nielsen class of A₅ and 3-cycles, but now use an arbitrary large number of them, has the Nielsen-RIGP been shown. A break through on this would likely, even for this case, along with the Main Conjecture of Modular Towers and nilpotent results, be major steps in the remainder of the RIGP.

REFERENCES:

[BaFr], P. Bailey and M. D. Fried, Hurwitz monodromy, spin separation and higher levels of a Modular Tower, Arithmetic fundamental groups and noncommutative algebra, PSPUM vol. 70 of AMS (2002), 79–220. arXiv:math.NT/0104289 v2 16 Jun 2005.

[CaDe] A. Cadoret and P. Debes, Abelian obstructions in inverse Galois theory, Manuscripta Mathematica (to appear).

[CaTa] A. Cadoret and A. Tamagawa, Uniform boundedness of p-primary torsion of Abelian Schemes, preprint as of June 2008.

[DeDes] P. Dèbes and B. Deschamps, The Inverse Galois problem over large fields, in Geometric Galois Action, London Math. Soc. Lecture Note Series 243, L. Schneps and P. Lochak ed., Cambridge University Press, (1997), 119–138.

[DeFr] P. Dèbes and M. D. Fried, Nonrigid situations in constructive Galois theory, Pacific Journal 163 (1994), 81–122.

[Fr1] M. D. Fried, Introduction to Modular Towers: Generalizing dihedral group–modular curve connections, Recent Developments in the Inverse Galois Problem, Cont. Math., proceedings of AMS-NSF Summer Conference 1994, Seattle 186 (1995), 111–171.

[Fr2] M. D. Fried, Enhanced review of J.-P. Serre's Topics in Galois Theory, with examples illustrating braid rigidity, Recent Developments in the Galois Problem, Cont. Math., proceedings of AMS-NSF Summer Conference, Seattle 186 (1995), 15–32.

[Fr3] M. D. Fried, Alternating groups and moduli space lifting Invariants, description and properties of spaces of 3-cycle covers, Arxiv #0611591. 01/04/09 To appear in Israel J.

[FrJ] M. D. Fried and M. Jarden, Field Arithmetic, Springer Ergebnisse II Vol 11 (1986) 455 pgs., 2nd edition 2004, 780 pps. ISBN 3-540-22811-x. We quote the second ed., but all references are to statements also in the first.

[FrVo] M. D. Fried and H. Völklein, The embedding problem over a Hilbertian PAC-field, Ann. Math. 135, (1992), 469–481.

[Hi] D. Hilbert, Über die Irreduzibilität ganzer rationaler Funktionen mit ganzzahligen Koeffizienten, J. Crelle 110 (1892), 104–129 (=Ges. Abh. II, 264–286).

[MM] G. Malle and B.H. Matzat, Inverse Galois Theory, Springer 1999, Monographs in Mathematics, ISBN 3-540-62890-8.

[Me] J.-F. Mestre, Extensions règuliéres de Q(t) de groupe de Galois ^{^}A_n, J. of Alg. 131 (1990), 483–495.

[P] B. Poonen, Computational aspects of curves of genus at least 2, 283–306 in Cohen, H. ed., Algorithmic Number Theory, Second International Symposium, ANTS-II, Talence, France, May 1996, Proceedings, Lecture Notes in Computer Science 1122, Springer-Verlag, Berlin. [MR 98c:11059].

[Po] F. Pop, Embedding problems over large fields, Ann. Math. 144 (1996), 1–35.

[Se] J.-P. Serre, Topics in Galois Theory, 1992, Bartlett and Jones Publishers, ISBN 0-86720-210-6, BAMS 30 #1 (1994), 124–135.

[Vo] H. Völklein, Groups as Galois Groups, an Introduction, Cambridge Studies in Mathematics 1996 53, ISBN 0-521-56280-5.

Pierre Debes and Mike Fried, Monday, February 9, 2009