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

This exposition relates four famous problems, denoted below by I(nverse) G(alois) P(roblem), RIGP, 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. None is known for any number field.

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. We 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 is an L/F with G=G(L/F).

The profinite group of all automorphisms of F– fixed on F is the absolute group GF. 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 GF.

RIGP FORMULATION: 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.

RIGP USE: 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 popup.

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 examplars (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:
  1. Interpretation of it as a problem of finding Q solutions of well-defined equations.
  2. Its relation to Hilbert's Irreducibility Theorem.
  3. Effective methods for solving it in cases.
  4. 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 in Modular Towers.

REGULAR SPLIT EMBEDDING PROBLEM (RSEP): This conjecture from [DeDes] plays a central role in Inverse Galois Theory:

Conjecture: 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:

  1. The fixed field of ker(f) in F is E; and
  2. F ∩ K– = E ∩ K–
#1 defines a proper solution to the embedding problem (E/K(z); f) over K(z). #2 is termed the condition for a regular solution. So, an alternate formulation using the language of this embedding problem for the pair (G,H) asks this. Does every split finite embedding problem for GK(z)  →G(E/K(z))=G have a proper regular solution mapping to H. 

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  Qab – 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 GK is a projective (profinite) group (see [FrJ, Chap. 22] or the definition p-Frattini-cov). The RSEP implies a more general conjecture of [FrVo]: If K is a Hilbertian countable field with GK projective, then GK is profree. A P(seudo)A(lgebraically)C(losed) field is automatically projective [FrJ, Thm. 11.6.2] (originally due to Ax). So, the Main result of [FrVo], that the conclusion of this conjecture holds for PAC Hilbertian fields, is a special case.

The argument has three steps [DeDes]:
The Shafarevich conjecture follows as Qab 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 EP Conjecture was proved by Pop [Po] for K a large field: Every smooth absolutely irreducible curve of K has infinitely many K points provided it has one. Another topic deserves mention here.

BECKMANN-BLACK PROBLEM: given a finite Galois extension E/K with group G, there exists a regular Galois extension F/K(z) with group G that specializes to E/K at some unramified point z0. This problem has the virtue of raising the relation between the RIGP and IGP in the sense of considering how regular realizations may be as plentiful as ordinary realizations.

REFERENCES:

[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.

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


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

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


[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 01/28/07