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
Q–.
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).
PROFINITE GROUP THEORY:
FINITE FIELD CHAPTERS:
GENERALIZATION OF FINITE
FIELDS TO P(seudo)-A(lgebraically)-C(losed) FIELDS:
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:
ELEMENTS OF ALGEBRAIC
GEOMETRY:
ELEMENTS OF DECISION
PROCEDURES:
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 you 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:
- 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).
- 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 Qpof 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) and he listed the exceptional p to this group being simple. That
means: 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, as 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 as 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," for a strikingly well defined
collection of fields F consider
"F ⇒ equations E over Q with solutions 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 simplify
their relation. On one hand, all absolutely irreducible equations
over such fields have rational points. On the other, an equation
satisfying HIT is one that doesn't have
too many solutions. While PAC fields have no nontrivial
valuations – often what a number theorist considers as arithmetic –
variants on HIT do define completely different sets of PAC fields. An
archetype result considers PAC subfields of Q– that
satisfy HIT: They automatically have pro-free absolute Galois group;
and 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 P1HIT
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 Q–
galois over Q is
Q– 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 gives a model for how diophantine properties of fields gave
meaning to Field Arithmetic. The book's structure in the sections below
followis 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 this author, it is an additional blessing that classical
connections – should you care about them – are not lost, but given
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.
I. CHAPTERS SUPPORTING THE MAJOR THEMES:
PROFINITE GROUP THEORY:
back-to-top
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
FINITE FIELD CHAPTERS:
back-to-top
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
II. BACKGROUND CHAPTERS
ELEMENTS OF ALGEBRAIC
GEOMETRY:
back-to-top
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
ELEMENTS OF DECISION
PROCEDURES:
back-to-top
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
Q–,
American J. of Math 100
(1978), 653–666.
[Fr] M. Fried, Galois
groups and Complex Multiplication, Trans.A.M.S. 235 (1978), 141--162.
[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 mfried@math.uci.edu