Field Arithmetic

The usual Galois correspondence between subgroups of Galois groups of finite Galois extensions and intermediate fields is not valid for infinite Galois extensions. The Krull topology restores this correspondence for closed subgroups (Proposition 1.3.1).

This chapter introduces the basic elements of the theory of valuations, especially discrete valuations, and of Dedekind domains. These sections are primarily a survey. We prove that an overring of a Dedekind domain is again a Dedekind domain (Proposition 2.5.7).

This chapter centers around the notion of linear disjointness of fields. We use this notion to define separable, regular, and primary extensions of fields. In particular, we prove that an extension ??/?? with a ??-rational place is regular. Section 3.7 gives a useful criterion for separability with derivatives.

Sections 4.1–4.4 survey the theory of functions of one variable; the Riemann–Roch Theorem; properties of holomorphy rings of function fields; and extensions of the field of constants.

In this chapter k is a finite field of characteristic p with q elements. Let F be an algebraic function field of one variable over K and g the genus of F/K.

The estimate on the number of prime divisors of degree 1 of a function field ?? over F?? (Theorem 5.5.2) leads in this chapter to an estimate on the number ?? of ??-rational zeros of an absolutely irreducible polynomial ?? ∈ F?? [??,??].

The major connection between the theory of finite fields and the arithmetic of number fields and function fields is the Chebotarev density theorem. Explicit decision procedures and transfer principles of Chapters 23 and 35 depend on the theorem or some analogs. In the function field case our proof, using the Riemann hypothesis for curves, is complete and elementary. In particular, we make no use of the theory of analytic functions. The number field case, however, uses an asymptotic formula for the number of ideals in an ideal class, and only simple properties of analytic functions. In particular, we do not use Artin’s reciprocity law (or any equivalent formulation of class field theory). This proof is close to Chebotarev’s original field crossing argument, which gave a proof of a piece of Artin’s reciprocity law for cyclotomic extensions.

We develop the basic concepts of logic and model theory required for applications to field theory.

The later chapters of this book give decision procedures for various theories of fields. If T denotes such a theory, we might give a naive description of a decision procedure as a system of instructions for deciding in a finite number of steps whether or not a given sentence 𝜃 belongs to T. We elaborate, more precisely, on two types of procedures, model-theoretic and algebraic.

We establish a simple algebraic elimination of quantifiers procedure for the theory of algebraically closed fields. This theory is model complete (Corollary 10.3.2). Among the applications are Hilbert’s Nullstellensatz and the Bertini–Noether theorem.

Here we present the algebraic geometry background for the study of PAC fields. The central result is a descent argument which associates to each variety V defined over a finite extension L of a field ?? a variety ?? defined over ??. Throughout this chapter and subsequent chapters we make the following convention: Whenever we are given a collection of field extensions of a given field ??, we assume that all of them are contained in a common field.

By Hilbert’s Nullstellensatz, algebraically closed fields are PAC. So are separably closed fields [Lan64, p. 76, Prop. 10]. Both statements are also immediate consequences of Theorem 12.2.3 below. We shall see many nonseparably closed PAC fields.

Various alternative proofs of the irreducibility theorem apply to other fields (including all infinite finitely generated fields). We call them Hilbertian fields.

Global fields and functions fields of several variables have been known to be Hilbertian for three quarters of a century. These are the “classical Hilbertian fields”.

Each finite proper separable extension of a Galois extension of a Hilbertian field K is Hilbertian. This is a theorem of Weissauer. Moreover, if L1 and L2 are Galois extensions of ?? and neither of them contains the other, then L1 L2 is Hilbertian. The diamond theorem says even more: each extension M of K in L1 L2 which is contained in neither L1 nor in L2 is Hilbertian (Theorem 15.2.3). An essential tool in the proof is the twisted wreath product (Section 15.1).

A. Robinson invented “nonstandard” methods in order to supplement the Weierstrass 𝜀, 𝛿 formalism of the calculus by a rigorous version of the classical calculus of infinitesimals in the spirit of Leibniz and other formalists. We will apply the nonstandard approach to algebra in the next chapter in order to find new Hilbertian fields. Its main virtue, from an algebraic point of view, is that it creates additional algebraic structures to which well-known theorems can be applied.

One may reapproach the classical Hilbertian fields through Proposition 17.1.1 [Roq75b], but it is easier to apply the following weaker condition of Weissauer.

Given a field K, one may ask which finite groups occur as Galois groups over K. If K is Hilbertian, then every finite group that occurs over K(t), with t transcendental over K, also occurs over K.

A finitely generated profinite group G has for each positive integer ?? only finitely many open subgroups of index at most n (Lemma 19.1.2). Groups that satisfy the latter condition are “small”. Small groups have in the category of profinite groups a property similar to the one that finite sets have in the category of all sets: Every epimorphism of a small profinite group onto itself is an isomorphism (Proposition 19.1.6(a)).

We continue the discussion on profinite groups of Chapters 1, 18, and 19. Central to this chapter is a discussion on free profinite groups. In particular, we prove that an open subgroup of a free profinite group is free (Proposition 20.6.2).

It is well known that every locally compact group admits a (one sided) translation invariant Haar measure. Applications of the Haar measure in algebraic number theory to local fields and adelic groups appear in [CaF67, Chap. II] and [Wei67]. Here we use it to investigate absolute Galois groups of fields. Since these groups are compact, the Haar measure is a two-sided invariant. We provide a simple direct proof of the existence and uniqueness of the Haar measure of profinite groups (Sections 21.1 and 21.2).

Since one of the goals of this book is to present decision procedures for theories of PAC fields, we require a preliminary concept, “a presented field with elimination theory”, in order to display the essential (for our purposes), basic explicit operations of field theory and algebraic geometry. In particular, all fields finitely generated over their prime fields, and their algebraic closures, are fields with elimination theory (Corollary 22.2.10).

This chapter presents one of the highlights of this book, the study of the elementary theory of 𝑒-free PAC fields. We apply the elementary equivalence theorem for arbitrary PAC fields (Theorem 23.3.3) to the theory of perfect 𝑒-free PAC fields containing a fixed countable base field 𝐾.

This chapter includes the theory of Ci-fields, Kronecker conjugacy of global field extensions, Davenport’s problem on value sets of polynomials over finite fields, a solution of Schur’s conjecture on permutation polynomials, and a solution of the generalized Carlitz conjecture on the degree of a permutation polynomial in characteristic 𝑝. Each of these concrete problems focuses our attention on rich historically motivated concepts that could be overlooked in an abstract model-theoretic viewpoint.

In this chapter we define a projective group as a profinite group 𝐺 for which every embedding problem is weakly solvable. By Gruenberg’s theorem (Lemma 25.3.2), it suffices to weakly solve only finite embedding problems. This leads to the second characterization of projective groups as those profinite groups which are isomorphic to closed subgroups of free profinite groups (Corollary 25.4.6).

The absolute Galois group of a PAC field is projective (Theorem 12.6.2). But not every field with projective absolute Galois group is PAC (Example 26.1.4).

The embedding property (Proposition 20.7.4) for free profinite groups is essential to the primitive recursive procedure for perfect PAC fields with free absolute Galois groups (Theorem 34.6.2). Since this is only a special case of a general result, we focus attention here on PAC fields whose absolute Galois groups have the embedding property: the Frobenius fields. The “field crossing argument” (e.g., the proofs of Proposition 7.4.8, Part G in Section 7.5, Part C of Proposition 18.8.6, Lemma 23.2.2, and Lemma 26.2.1) applies to give an analog, for Frobenius fields, of the Chebotarev density theorem (Proposition 27.1.4).

In order to characterize profinite groups of rank exceeding ℵ0 by their finite quotients, in addition to the embedding property, we must add an hypothesis concerning the cardinality of the set of solutions of finite embedding problems.

The expression in the braces approaches the inverse of the basis of the natural logarithms (which is usually denoted by the same letter 𝑒 which we are using here for another purpose). The exponent of the braces approaches infinity as ?? approaches infinity.

Let 𝐾 be a countable Hilbertian field. By Proposition 13.3.3, each finite extension of 𝐾 is Hilbertian.

For a more compelling example, which unfortunately partially relies on material outside the scope of this book, consider a prime number 𝑝.

In contrast to the theories considered so far (e.g. the theory of finite fields, the theory of almost all fields ˜Q (𝜎1, . . . , 𝜎e) for fixed ??, and the theory of perfect PAC fields of bounded corank), the theory of perfect PAC fields is undecidable. This is the main result of this chapter (Corollary 32.10.2). An application of Cantor’s diagonalization process to Turing machines shows that certain families of Turing machines are nonrecursive. An interpretation of these machines in the theory of graphs shows the latter theory to be undecidable. Finally, Frattini covers interpret the theory of graphs in the theory of fields. This applies to demonstrate the undecidability of the theory of perfect PAC fields.

Throughout this chapter we work over a fixed infinite base field ??, finitely generated over its prime field. By Theorem 14.4.2, K is Hilbertian.

Chapter 34 extends the constructive field theory and algebraic geometry of Chapter 22, in contrast to Chapter 23, to give effective decision procedures through elimination of quantifiers.

Chapter 34 establishes the Galois stratification procedure over a fixed field K with elimination theory. The outcome is an explicit decision procedure for the theory of perfect Frobenius fields that contain K.

The first section of this chapter lists those problems of the first three editions of “Field Arithmetic”F that have been solved or partially solved. The second section contains open problems.