The Domain of Hilbert's Irreducibility Theorem

The phrase HIT result will refer to any statement over a field that captures or enhances the original statement:

(*) Given an irreducible polynomial f(z,w) with coefficients in Q in two variables of degree n in w, then for infinitely many z0 in Q (even in the integers Zf(z0,w) is irreducible in  w of degree n over Q. I use Q- for the algebraic closure of Q.

To understand the domain of the result, we notice first that it translates to a statement about projective normal (nonsingular in this case) covers. I explain that. The set in the complex numbers defined by X'={(z,w) | f(z,w)=0} maps to the z-line P1z by φ': (z,w) → z. The degree of this map – number of points in Q- in the fiber over all but finitely many z0 in Q- is also n. A singular point (z0,w0)∈ X' is one at which the partials of f in both z and w are zero. Denote X' with its singular points removed by X''. There is then a unique non-singular projective algebraic variety X containing X'' as a natural subset and having a map (function) φ: X P1z with these properties:
  1. φ extends φ' on X'.
  2. φ is a finite flat morphism – of degree n in every fiber.
Since only a finite number of points have changed, only the wording of statement (*) changes to say that there are infinitely many z0 in Z for which the fiber is irreducible over Q. Denote the set of exceptions to HIT for (φ, X) by Rφ, its complement in Z by Iφ. There are many advantages to changing (φ', X') to (φ, X). For one, putting a polynomial f in front of someone gives them nothing but the coefficients of the polynomial. There is no useful "name" or "type" attached to the polynomial.

There are, however, nice pieces of data attached to the type of (φ, X). In asking to enhance (*) we can make use of their "type," called a Nielsen class. The simplest observation on a cover is that it comes with two groups, its arithmetic (resp. geometric) Galois closure group  ˆGφ (resp. Gφ).  Hilbert's application of his HIT was to show for a given (φ, X), for infinitely many z0 in Q, the specialized polynomial f(z0,w) also has ˆGφ as its Galois closure group. Call the set of such z0 in Z the Galois closure set GCφ. When we want to consider values of z0 in Q, or over another domain, we add corresponding indicative decoration. Now I list a collection of enhancements; most are still centers of active research now.

After a listing of our major topics, there is a URL on that topic. First there are results in the classical domain, over any number field, especially Q.
  1. HIT applies to specialize general objects defined by covers to specific cases. Applications like specializations of the Galois closure of a cover and Néron's production of elliptic curves of high rank over Q simultaneously applies HIT to many covers. HITcovers.html
  2. For infinite collections of polynomials f(z,w) based on their type, we can describe the set Rφ, up to a finite set: A near description of Rφ. Example: Applying HIT to the Inverse Galois Problem, have simplifications using Nielsen classes, allowing a near description of GCφ. HITsiegel.html
  3. For any Hilbertian set Iφ finding an explicit representative in it is a tough problem. Subtler questions arise for (infinite) Universal Hilbert Subsets: sets nearly contained, meaning up to a finite set, in each Rφ. HITUHS.html
  4. There are infinite sequences of covers … → XkXk-1 →  … → X0  →  P1z, a tower, for which an enhancement of statement (*) holds: (**) For infinitely many z0 in Q the fiber of Xk →  P1z over z0  is irreducible for each k.  A most significant phrase, Frattini cover. HITOIT.html
The accompanying files on #3 and #4 start with an archetype that will help the reader accustom to covers. The distinction between ˆGφ and Gφ in #4 show how to use a geometric expression for describing GCφ. #3 also raises a practical consideration of whether there is an HIT result by replacing P1z by other spaces. There is! Still, the depth of it would be to specialize fibers over certain special rational points when the base of the cover is group scheme.

All methods of transparently finding a representative of Iφ, such as finding an explicit arithmetic progression in it, run into nontrivial points about explicit primes. Debes and Zannier, and their cowriters, have explored the connections of diophantine geometry conjectures to this topic. There is no hope of classifying Universal Hilbert Subsets in #5, but their existence first started as a general argument, and then led to making them explicit. Given one such, finding the last member not in a given Iφ must be a very subtle problem.

The archetype for #6 is a tower of modular curves; there  the precise result is called Serre's Open Image Theorem. By thinking of HIT in this way, we point out that there are myriad circumstances theoretically suiting #6. It is a major research project to prove results for them. A subfield L of Q- is called Hilbertian if a version of (*) holds for all two-variable polynomials over L.
  1. It is known that in the lattice of fields, the Hilbertian property jumps around all over the place. Yet, Weiesauer's Statement plays a special role in identifying Hilbertian fields. HITWeissauer.html
  2. In the class of P(seudo)A(lgebraically)C(losed), the Galois group of the field interprets HIT, and many distinct variants on it. HITPAC.html
  3. Those PAC fields that are Galois extensions of Q provide a way to describe the absolute Galois group of Q, GQ. HITGQ.html
Versions of HIT are compatible with various versions of other diophantine properties. Our archetype is from §3 of HIT74.pdf, giving an HIT version as a consequence of the Chebotarev density theorem.  By considering the HIT property simultaneously with other properties of fields, in varying the field, #7 and #8 open  relations among various diophantine properties, including HIT.  Various analogs of Hilbertianity, arise – our 2nd application of Frattini covers –. As an example, #8 compares the Fried-Jarden Σ-Hilbertian fields with various Corvaya-Zannier fields.

#9 is the territory of Fried-Voelklein: For PAC fields, Hilbertianity is equivalent to pro-free absolute Galois group. (It thereby produced what are still substantially the only known presentations of the absolute Galois group of Q, but it also shows how close is Fried-Voelklein to proving generalizations that include Shafarevich's conjecture.)  PAC fields and, more generally, fields of projective dimension 0 (e.g., cyclotomic numbers) are tools for dissecting GQ.

A theme in the Fried-Jarden book Field Arithmetic, is to collect subfields of the algebraic numbers according to various possible of their diophantine properties. By looking at fields satisfying subsets of properties, one sees the intrinsic relation between the properties. With the Fried-Voelklein result describing presentations of the absolute Galois group of Q, we see that properties of large fields can reveal structure in classical objects. The main tool in this is a variant of Hilbert's Theorem. Likewise, a view of Serre's Open Image Theorem – on modular curves – is that towers of covers can exhibit a Hilbertian property. Those two topics are deeper results. Yet, in this file we see that variant's on Hilbert's Theorem make clever tools for investigating many problems.

Moshe Jarden didn't use my writing on the bigger themes that relate to the Hilbertian property. They may have been too abstract for his tastes. Still, relating HIT to other diophantine properties was underdone in Fried-Jarden. Especially since that was what attracted Jarden to the book idea when I first proposed the title "Field Arithmetic" and its major topics in 1975. It was I who refereed Jarden's thesis (James Ax gave it to me) and solved the problems his thesis had left unsolved (in my Math Reviews of the same paper).

January 12, 2009 Michael D. Fried