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 z

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 P

- φ extends φ' on X'.
- φ is a finite flat morphism – of degree n in every fiber.

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 ˆG

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.

- 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

- For infinite collections of polynomials f(z,w) based on their type, we can
describe the set R
_{φ}, up to a finite set: A near description of R_{φ}. Example: Applying HIT to the Inverse Galois Problem, have simplifications using Nielsen classes, allowing a near description of GC_{φ}. HITsiegel.html

- For any
*Hilbertian*set*I*_{φ}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 R_{φ}. HITUHS.html

- There are infinite sequences of covers … → X
_{k}→ X_{k-1}→ … → X_{0}→ P^{1}_{z}, a tower, for which an enhancement of statement (*) holds: (**) For infinitely many z_{0}in Q the fiber of X_{k}→ P^{1}_{z}over z_{0}is irreducible for each k. A most significant phrase, Frattini cover. HITOIT.html

All methods of transparently finding a representative of

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

- 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

- 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
- Those PAC fields that are Galois extensions of Q provide a way to describe the
absolute Galois group of Q, G
_{Q}. HITGQ.html

#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 G

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