Field Arithmetic now has its own classification in Mathematical Reviews: 12E30 Field Arithmetic. Admittedly the new Springer cover for the book is very attractive. Still, what else recommends "Field Arithmetic" as a subject of study, and why would it make sense for two such different mathematicians to put together that study? To help understand that I have below divided the 31 chapters (not counting a collection chapter: Chapter 32. Problems of Field Arithmetic . . . . . . . . . . . 748) into subcollections. We denote the rational numbers by Q, and the algebraic numbers by Qcl. This discourse is related to that on the Regular Inverse Galois Problem in RIGP

The major collections of modern material is under I. CHAPTERS SUPPORTING THE MAJOR THEMES: where you will find these subcategories (on which you can click: return here by clicking the back-to-top element).

HILBERTIAN (and variant) FIELDS:

The Table-of-Contents has detailed listing of subchapters. These include a listing for the problems sets – most of which come from the first edition.

Some chapter listings below include an addition called Highlights. Many indicate discussions not in the first editon.  By highlights I mean subthemes and results that help define Field Arithmetic. The book, however, is meant to include much old background, often disconnected in the literature. By making the book so self-contained, the authors were bucking a trend. Further, sometimes that insistence on considerable filled-detail makes a bumpy ride. My heading below for chapters meant to take from an older literature is II. BACKGROUND CHAPTERS. Under it you will find:


Now I return to the theme supported by the major categories. A field is a domain in which you can do traditional arithmetic: adding, subtracting, multiplying, and dividing – except where a 0 might appear in the denominator. You would do such things in order to solve equations. The authors of this book have had – at times – diophantine interests. That means two related things:
  1. Finding equations for which a solution corresponds to an object of interest. Like, a point on a modular curve corresponds to an elliptic curve (an equation in its own right) with something extra on it (usually a collection of division points).
  2. A refined idea of what you want to understand about solutions of collections of equations. Like, which among a set of modular curves has an infinite number of solutions over Q, or over p-adic completions, Qp, of Q.
HISTORICAL MOTIVATION FROM MODULAR CURVES: Galois Theory is the tool par-excellence for translating naive/general qualitative problems into equations. It is an interpretive art – rather than an easily learned procedure – still, over 177 years after the death of Galois. One reason for that – despite the worship of Galois and his theory – is most elementary textbook authors are enamoured of the group theory, not the interpretive aspects,  around it. Both authors were influenced by other researchers who cared deeply about diophantine matters.

Our short list with chapters which discuss problems – whose solutions fall under Field Arithmetic solutions – started with them: Ax (Ch. 11, 18, 21, 31), Davenport and Lewis (Ch. 21), Roquette (Ch. 15, 18), Siegel (Ch. 21). Books with similar historical motivation are [Se1] and [Se2], though these are more overt about using equations (especially elliptic and modular). They also use much more advanced techniques – while Field Arithmetic stays within techniques developed in the book. All three books regard HIT (see below) as a general result whose variants are potentially general tools (as in [Se1, Chap. 9] and [Se2, Introduction]). The idea already appears in Siegel [Si] and it seriously motivated Néron [Ne].

It isn't fanciful to raise Galois and modular equations close to each other, as documented by Rigetelli [Ri]. After Abel's introduction of the elementary modular curves called X0(p),  Galois showed their equations were unsolvable by the standard of the day. He explained this to mean the natural cover X0(p) →  P1 has geometric monodromy group SL2(p), a simple group for all but finitely many p, and he listed the exceptional p to this group being simple. That means, excluding the exceptional p: You cannot express uniformizing variables for them as radicals involving the classical j-invariant. That italicized text is a specific interpretation of using known functions.

THE ANGUISH OVER DECIDABILITY: [Ri] also convincingly documents the case for Galois being a suicide – of an arranged kind, related to political matters – over several disappointments. These included his rejection by the mathematical establishment (prematurely at 20, I would think! – but not getting into the Harvard of its day may also have exacerbated this), and a young lady (much classier than the tart of standard lore). Further, [Ri] suggests Galois' father's suicide had a terribly profound effect on him. So did a time in prison related to the failure of his own political goals. 

Trying to solve complicated equations reveals practical aspects of a theoretical difficulty.  Rarely can one solve them in one important sense. Though solutions may exist they will not be related to functions studied previously.  Rather than solutions, however, most scientists want properties of solutions.  Many of the papers in paplist-cov use group representation theory to avoid solving equations. This is the monodromy method, and Field Arithmetic has many introductory aspects to that subject.

Yet, our present day notions of (un)decidability are more sophisticated. Field Arithmetic has many examples of decidability for which the book may be a rare source for diophantine motivated researchers. It is profinite group theory that allows precise results.

PROFINITE APPEARANCE OF EQUATIONS: Quite profoundly, equations – in imitation of modular curves – tend to come in profinite collections. Deligne [De] first raised this for modular curves in lectures at IAS in 1972. [Fr, Sect 4] gave meaning at the time to this version: Modular curves were profinite equations coming from dihedral groups.  It wasn't until I had the universal Frattini cover that I saw how to generalize that connection between modular curves and dihedral groups to an association that attached to each finite group a much more general profinite system of equations.

The theory of M(odular) T(owers) (to which the papers of paplist-mt are dedicated) has a general goal.  Showing the more general systems have modular curve-like properties. While MTs requires considerable algebraic geometry – and being a generalization of modular curves, some aspects of moduli – there are more elementary examples of this theme in Field Arithmetic.

SWITCH FIELDS WITH EQUATIONS: Field Arithmetic tests the theory of equations by considering collections of equations over fields more general than number fields. The idea: By considering fields of certain type, we can study the collection of equations over them with solutions of a particular type. So, instead of "one equation E ⇒ fields over which it has solutions," we switch the position of the field and the equation to consider this. For a collection of fields F – defined by a set of, possibly diophantine or Galois theorey properties – consider " F  ⇒ equations E (resp. 1st-order definable sets) over Q with solutions (resp. elimination of quantifiers) in F." The major example of this theme comes from the interplay of two diophantine properties equations might have: Absolute irreducibility and (satisfying) Hilbert's Irreducibility Theorem (HIT). An example is the switch from saying "HIT holds over Q" to considering the collection of Hilbertian fields. 

PAC FIELDS AS A MODEL: PAC fields are an example that helps understand this relation. We also learn from them the relation between different properties. The defining property of PAC fields is that all absolutely irreducible equations over such fields have rational points.

By contrast, an equation satisfying HIT is one that doesn't have too many solutions. While PAC fields have no nontrivial valuations – some number theorists consider that as a central subject of arithmetic – variants on HIT do define completely different sets of PAC fields. An archetype result considers PAC subfields of the algebraic closure, Qcl, of  Q that satisfy HIT: They automatically have pro-free absolute Galois group.  Likely so does any  field satisfying HIT having projective absolute Galois group [FrV].

Yet, there are PAC fields that satisfy R(egular) HIT – for each Galois regular cover of P1 HIT holds – that are not Hilbertian. An example use of the universal Frattini cover. A significant early collaboration of the authors was over a conjecture of Ax: The only PAC subfield of Qcl, Galois over Q is Qcl  itself.

Not only was that wrong [FrJ], but there are very classical looking PAC fields (like the totally real numbers with √ -1 = i adjoined). Yet, Ax also made an astonishing conjecture that turned out true: forms of degree d in projective d-space have points over any PAC field (see [Ax1], [Se3, §3.3 on C1fields]). From the finite field analogy, and Chevalley's Th., Ax divined a fundamental property of absolute irreducibility: Kollár has shown that a degree d hypersurface over Q in projective d space contains some absolutely irreducible Q subvariety [Ko].

SUMMARY AND CLASSICAL CONNECTIONS: The above models how diophantine properties of fields gave meaning to Field Arithmetic. The book's structure in the sections below follows this chain. If we can use the structure of profinite groups handily (see PGT), it is possible to see equations akin to those over finite fields in a more general way (see FFC). Then, connecting the domains of the Cebotarev Density Theorems, HIT and the Riemann Hypothesis (see HF) leads us to view diophantine properties as defining collections of fields (arithmetic domains; see GFF).

For me, it is an additional blessing that classical connections – should you care about them – are not lost, but give additional power. [Ax2] in the late '60s saw a way to use finite field thinking and logic in general diophantine problems that influenced Jarden as in [J]. The central topic of the Cebotarev Density Theorem, expanded the idea of reducing general diophantine questions with quantified variables to special types through Galois stratification (see [annals76.pdf]). This allowed Chow motives over Q to substitute for questions viewed over all completions of number fields  ([DLo], [Ha] and [Ni]).  We expect (as in exceptTowYFFTA_519.pdf §8.2.2]) to see an elementary approach to equivalences on Chow motives to be one consequence.

When  [annals76.pdf] paper was published in 1976, there was a reaction of model theorists, to its use of schemes, for example. Yet, there was a deeper reaction at the time. Said they, "What is the difference between primitive recursive and just plain recursive?" After all – they claimed – neither is truly effective. But it was already built into the published paper on Galois stratification that this was wrong, when it used Davenport's problem to illustrate how Galois stratifications are a generalization of the easiest part of the solution of that problem. I used it because the problem had cachet before Ax's work motivated Galois stratification. Also, the complete solution of Davenport's problem had several effects on the classical mathematical community.

For example, in going beyond the domain of 1st order statements, it conjectured results established only with the completion of the classification of finite simple groups. Galois stratification was much more than primitive recursive. That first preliminary step was limited because it applied only to families of pairs of polynomial covers (P1 → P1) at a time, not allowing the degree to be a variable, too. Yet, there was an inspiration from just applying the first step of the Galois formula reformulation. In recasting all the papers over the years that used a piece of [Fr1], in [Fr3] we see that Galois formulas enabled sophisticated group theory previously unguessed at. 


Chapter 1. Infinite Galois Theory and Profinite Groups . . . . . . 1
Chapter 17. Free Profinite Groups . . . . . . . . . . . . . . . 337
    Highlight: Categories of profinite groups where there exist free objects.
Chapter 22. Projective Groups and Frattini Covers . . . . . . . 494
    Highlight: Construction and properties of the universal Frattini cover of a finite group.
Chapter 23. PAC Fields and Projective Absolute Galois Groups . . 541
    Highlight: The absolute Galois group of a PAC field is projective, allowing production of many groups as absolute Galois groups.
Chapter 25. Free Profinite Groups of Infinite Rank . . . . . . . 591
    Highlight: Haran's generalization of Weissauer; properties of absolute Galois groups of Hilbertian Fields.
Chapter 26. Random Elements in Free Profinite Groups . . . . . 632
    Highlight: Description of the closed subgroup and closed normal subgroup generated by e random elements the absolute Galois group of a Hilbertian field.
Chapter 27. Omega-Free PAC Fields . . . . . . . . . . . . . . 652

Chapter 4. The Riemann Hypothesis for Function Fields . . . . . 77
    Highlight: Bombieri's proof in detail.
Chapter 6. The Chebotarev Density Theorem . . . . . . . . . . 107
    Highlight: The tool most generalized to deal with general diophantine equations.
Chapter 21. Problems of Arithmetical Geometry . . . . . . . . 452
    Highlight: Schur, Davenport and Ci field problems  motivate explicit use of quantifier elimination; using serious group theory. 
Chapter 31. Galois Stratification over Finite Fields . . . . . . . 727
    Highlight: By generalizing elementary statements to Galois stratifications, you can eliminate quantifiers.

GENERALIZATION OF FINITE FIELDS TO P(seudo)-A(lgebraically)-C(losed) FIELDS: back-to-top
Chapter 11. Pseudo Algebraically Closed Fields . . . . . . . . . 192
    Highlight: The completion with respect to any valuation of a PAC field K is dense in its separable closure.
Chapter 18. The Haar Measure [on absolute Galois groups] . . . . . . . . . . . . . . . . 362 (and finding PAC subfields of the algebraic numbers)
    Highlight: PAC, Hilbertian fields in abundance as Galois extensions of any Hilbertian field.
Chapter 20. The Elementary Theory of e-Free PAC Fields . . . . 427
    Highlight: Elimination of quantifiers in PAC fields in Abundance.
Chapter 24. Frobenius Fields . . . . . . . . . . . . . . . . . 559
    Highlight: PAC fields with Cebotarev properties and solution for PAC fields of the Beckmann-Black problem .
Chapter 29. Algebraically Closed Fields with Distinguished Automorphisms . . 695
Chapter 30. Galois Stratification . . . . . . . . . . . . . . . 705
    Highlight: Near rationality of Zeta functions attached to diophantine statements.

HILBERTIAN (and variant) FIELDS: back-to-top
Chapter 12. Hilbertian Fields . . . . . . . . . . . . . . . . . 218
Chapter 13. The Classical Hilbertian Fields . . . . . . . . . . . 230
Chapter 15. Nonstandard Approach to Hilbert's Irreducibility Theorem . . . . 276
Chapter 16. Galois Groups over Hilbertian Fields . . . . . . . . 290


Chapter 2. Valuations and Linear Disjointness . . . . . . . . . 19
Chapter 3. Algebraic Function Fields of One Variable . . . . . . 52
Chapter 5. Plane Curves . . . . . . . . . . . . . . . . . . . 95
Chapter 10. Elements of Algebraic Geometry . . . . . . . . . . 172
Chapter 19. Effective Field Theory and Algebraic Geometry . . . 401

Chapter 7. Ultraproducts . . . . . . . . . . . . . . . . . . . 132
Chapter 8. Decision Procedures . . . . . . . . . . . . . . . . 149
Chapter 9. Algebraically Closed Fields . . . . . . . . . . . . . 163
Chapter 14. Nonstandard Structures . . . . . . . . . . . . . . 266
Chapter 28. Undecidability . . . . . . . . . . . . . . . . . . 668
    Highlight: Undecidable theories from collections of PAC fields.

[Ax1] J. Ax, Proof of some conjectures on cohomological dimension, Proc. AMS 16 (1965), 1214–1221.
[Ax2] J. Ax, The elementary theory of finite fields, Ann. of Math 88 (1968), 239–271.
[De] P. Deligne and M. Rapaport, Les schémas de modules de courbes elliptiques, Lect. Notes in Math. 349, Springer-Verlag (1973) 143–316.
[DLo] J. Denef and F. Loeser, Definable sets, motives and p-adic integrals, J. Amer. Math. Soc. 14 (2001), 429-469.
[FrJ] M. Fried and M. Jarden, Diophantine properties of subfields of Qcl, American J. of Math 100 (1978), 653–666.
[Fr1] M. Fried, The field of definition of function fields and a problem in the reducibility of polynomials in two variables, Illinois Journal of Math. 17, (1973), 128–146.  
[Fr2] M. Fried, Galois groups and Complex Multiplication, Trans.A.M.S. 235  (1978), 141--162.
[Fr3] M. Fried, Variables separated equations: Strikingly different roles for the Branch Cycle Lemma and the Finite Simple Group Classification, (DOI 10.1007/s11425-011-4324-4). Science China Mathematics, vol. 55, January 2012, 1–72
[FrV] M. Fried and H. Völklein, The embedding problem  over an Hilbertian-PAC field,  Annals of Math 135  (1992), 469–481.
[Ha] T. Hales, What is Motivic Measure?, BAMS vol. 42, Num. 2 (2005), 119–135 (last section added after print).
[J] M. Jarden, The elementary theory of ω-free Ax fields, Invent. math. 38 (1976), 187–206.
[Ko] J. Kollár, Algebraic varieties over PAC Fields, to appear in Israel J.
[Ne] A. Néron, Propriétés arithmétiques de ceertaines familles de courbes algébriques, Proc. Int. Coong. Amst. (1954), vol. III, 481–488.
[Ni] J. Nicaise, Relative motives and the theory of pseudo-finite fields, IRMN, to appear 2007.
[Ri] L.T. Rigatelli, Evariste Galois: 1811-1832, Vol. 11, translated from the Italian by John Denton, Vita Mathematica, Birkhäuser, 1996.
[Se1] J-P Serre, Aspects of Math., Translated and edited by M. Brown from notes of M. Waldschmidt, 3rd ed. 1997.
[Se2] J-P Serre, Topics in Galois Theory, 1992, Bartlett and Jones Publishers, BAMS 30 #1 (1994), 124–135. ISBN 0-86720-210-6.
[Se3] J-P Serre, Galois Cohomology, translated from the French by Patrick Ion, Springer 1997.
[Si] C.L. Siegel, Über einige Anwendungen diophantischer Approximationen, Abh. Pr. Akad. Wiss (1929), no. 1.

Mike Fried 02/05/07 (updated 03/16/13)