Field Arithmetic now has its own classification in Mathematical
Admittedly the new Springer
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
This discourse is related to that on the Regular Inverse Galois Problem
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
PROFINITE GROUP THEORY:
FINITE FIELD CHAPTERS:
FIELDS TO P(seudo)-A(lgebraically)-C(losed) FIELDS:
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
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:
CURVES: Galois Theory is the tool par-excellence for
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.
- Finding equations for which a solution corresponds to an
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
collections of equations. Like, which among a set of modular curves has
an infinite number of solutions over Q, or over p-adic
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]
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].
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
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
That italicized text is a specific interpretation of using known functions.
THE ANGUISH OVER
[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
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
properties of solutions. Many of the papers in paplist-cov use group
theory to avoid solving equations. This is the monodromy method,
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
for which the book may be a rare source for diophantine motivated
researchers. It is profinite group theory that allows precise results.
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]
at the time to this version: Modular curves were profinite equations
coming from dihedral groups. It wasn't until I had the universal Frattini
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
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
SWITCH FIELDS WITH
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
– defined by a set of, possibly diophantine or Galois
theorey properties – consider
⇒ 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
holds over Q"
the collection of Hilbertian fields.
PAC FIELDS AS A
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,
satisfy HIT: They automatically have pro-free absolute Galois
group. Likely so does any field satisfying HIT
absolute Galois group [FrV].
Yet, there are PAC fields that satisfy R(egular) HIT – for each Galois
regular cover of P1
holds – that are not Hilbertian. An example use of the universal Frattini
A significant early collaboration of the authors was over a conjecture
of Ax: The only PAC subfield of Qcl,
Galois over Q
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
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
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,
domains of the Cebotarev Density Theorems, HIT and the Riemann
us to view diophantine properties as defining collections of fields
(arithmetic domains; see
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
stratification (see [annals76.pdf]).
This allowed Chow motives over Q
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.
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
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 →
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
I. CHAPTERS SUPPORTING THE MAJOR THEMES:
Chapter 1. Infinite Galois Theory and Profinite Groups . . . . . . 1
Chapter 17. Free Profinite Groups . . . . . . . . . . . . . . . 337
profinite groups where
there exist free objects.
Groups and Frattini Covers . . . . . . . 494
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
Chapter 25. Free Profinite Groups of Infinite Rank . . . . . . . 591
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
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
Chapter 6. The Chebotarev Density Theorem . . . . . . . . . . 107
The tool most
generalized to deal
with general diophantine equations.
Chapter 21. Problems of Arithmetical Geometry . . . . . . . . 452
field problems motivate explicit use of quantifier
using serious group theory.
Chapter 31. Galois Stratification over Finite Fields . . . . . . . 727
statements to Galois stratifications, you can eliminate quantifiers.
FINITE FIELDS TO P(seudo)-A(lgebraically)-C(losed) FIELDS:
Chapter 11. Pseudo Algebraically Closed Fields . . . . . . . . . 192
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
fields in abundance
as Galois extensions of any Hilbertian field.
Chapter 20. The Elementary Theory of e-Free PAC Fields . . . . 427
quantifiers in PAC
fields in Abundance.
Chapter 24. Frobenius Fields . . . . . . . . . . . . . . . . . 559
PAC fields with
and solution for PAC fields of the Beckmann-Black
Chapter 29. Algebraically Closed Fields with Distinguished
Automorphisms . . 695
Galois Stratification . . . . . . . . . . . . . . . 705
of Zeta functions
attached to diophantine statements.
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
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
collections of PAC fields.
[Ax1] J. Ax, Proof
of some conjectures on cohomological dimension, Proc. AMS 16 (1965),
[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
[DLo] J. Denef and F. Loeser, Definable sets, motives and
J. Amer. Math. Soc. 14
[FrJ] M. Fried and M. Jarden, Diophantine properties of
American J. of Math 100
[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,
[Fr2] M. Fried, Galois
groups and Complex Multiplication, Trans.A.M.S. 235
[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
[FrV] M. Fried and H. Völklein, The embedding problem
Hilbertian-PAC field, Annals of Math 135
[Ha] T. Hales, What is Motivic Measure?,
(2005), 119–135 (last section
added after print).
[J] M. Jarden, The
elementary theory of ω-free
fields, Invent. math. 38
[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,
Galois: 1811-1832, Vol.
11, translated from the Italian by John Denton, Vita
[Se1] J-P Serre,
Aspects of Math.,
and edited by M. Brown from notes of M. Waldschmidt, 3rd ed. 1997.
[Se2] J-P Serre,
Topics in Galois Theory,
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
[Si] C.L. Siegel, Über
einige Anwendungen diophantischer Approximationen, Abh.
Akad. Wiss (1929), no. 1.
Mike Fried 02/05/07 (updated 03/16/13) firstname.lastname@example.org