UDC 519.48

MATHEMATICS

Yu. L. ERSHOV

ON FIELDS WITH A SOLVABLE THEORY

(Presented by Academician A. I. Mal'tsev on 20 VI 1966)

In the present note some new examples are indicated of fields and classes of fields of finite characteristic having a decidable elementary theory (for definitions connected with mathematical logic, see [1]).

1. Let \(p\) be a prime number, and let \(F\) be a field of characteristic \(p\). Put

\[ \tau(F)= \begin{cases} \log_p [F:F^p], & \text{if } [F:F^p] \text{ is finite},\\ \infty, & \text{if } [F:F^p] \text{ is infinite}. \end{cases} \]

A field \(F\) is separably closed if it has no proper algebraic separable extensions.

Proposition. Let \(F_1\) and \(F_2\) be separably closed fields of characteristic \(p\ne 0\). \(F_1\) and \(F_2\) are elementarily equivalent if and only if \(\tau(F_1)=\tau(F_2)\).

We shall give the proof only for the case \(\tau(F_1)=\tau(F_2)=\tau<\infty\). Let \(T\) be a system of axioms in the signature of the theory of fields with added symbols for constants \(a_1,\ldots,a_\tau\), satisfied only by models of the form \(\langle F,a_1,\ldots,a_\tau\rangle\), where \(F\) is a separably closed field of characteristic \(p\), \(\tau(F)=\tau\), and \(a_1,\ldots,a_\tau\) form a \(p\)-basis of \(F\) over \(F^p\) [5]. We shall show that the theory \(T\) is complete.

First, \(T\) is model complete. Indeed, let \(\langle F',a_1,\ldots,a_\tau\rangle \subset \langle F'',a_1,\ldots,a_\tau\rangle\) be two models of \(T\). Then \(F''\) is a regular extension of \(F'\) [2]. In fact, \(F''\) is separable over \(F'\), since \(a_1,\ldots,a_\tau\) remains a \(p\)-basis in \(F''\) over \(F''^p\); moreover, \(F'\) is separably closed and, consequently, algebraically closed in \(F''\). From the regularity of \(F''\) over \(F'\) and the separable closedness of \(F'\) it follows [2] that every point of \(F''\) has a specialization in \(F'\) over \(F'\). Hence the model completeness of \(T\) follows at once.

Second, among the models of the theory \(T\) there is a minimal one. Namely, consider the field \(Z_p(a_1,\ldots,a_\tau)\) of rational functions in \(\tau\) variables over the prime field \(Z_p\). The separable closure of this field is the minimal model.

From Robinson’s criterion [1] follows the completeness of the theory \(T\). This completes the proof in the case under consideration. The case \(\tau(F_1)=\infty\) is considered somewhat more complicatedly.

Theorem 1. The following classes of fields have a decidable theory:

a) the class of all separably closed fields;

b) the class of all separably closed fields of fixed characteristic;

c) the class consisting of one (arbitrary) separably closed field.

Theorem 1 may be regarded as a natural extension of A. Tarski’s result on the decidability of the theory of algebraically closed fields.

1. A finite field with \(p^r\) elements, where \(p\) is a prime number, will be denoted by \(F_{p,r}\). Let \(F\) be an absolutely algebraic field of characteristic \(p\),

i.e., the algebraic extension of the field \(Z_p\). With the field \(F\) we associate the following function \(\chi_F\):

\[ \chi_F(n)= \begin{cases} \max\limits_s\left(F_{p,p_n^s}\subset F\right)+1, & \text{if there exists an } s \text{ such that } F_{p,p_n^s}\not\subset F,\\ 0, & \text{otherwise,} \end{cases} \]

where \(p_n\) is the \(n\)-th prime number.

Let us note that for any two absolutely algebraic fields \(F'\) and \(F''\) of characteristic \(p\),

\[ F'=F''\Longleftrightarrow \chi_{F'}=\chi_{F''}, \]

and for any function \(\chi\) defined on \(\{0,1,\ldots\}\) and taking values in the same set, there exists an absolutely algebraic field \(F\) of characteristic \(p\) such that \(\chi=\chi_F\).

Theorem 2. Let \(F\) be an absolutely algebraic field of characteristic \(p\ne0\). Then \(F\) has a decidable theory if and only if \(\chi_F\) is a recursive function.

Theorem 3. Let \(F_1,F_2,\ldots\) be a sequence of finite fields such that: 1) \(F_i\subset F_{i+1}\); 2) the set of cardinalities \(\{\overline{F_i}\}\) is recursive; 3) if \(F_\infty=\bigcup F_i\), then \(\chi_{F_\infty}\) is a recursive function not taking the value \(0\). Then the class \(\{F_1,F_2,\ldots\}\) has a decidable theory.

The proofs of Theorems 2 and 3 use the technique of model completeness and an important result of Lang and Weil (3).

For the proof, one writes down a recursive system of axioms which assert that, if a system of polynomials defines an absolutely irreducible variety (see (4)) and the field contains sufficiently many elements (the estimate from (3) is used), then this variety has a rational point.

The second group of axioms consists of axioms asserting the existence or nonexistence of subfields of the form \(F_{p,p_n^s}\) (the existence or nonexistence of subfields of this kind in the presence of a sufficient number of elements), and here the function \(\chi_F(\chi_{F_\infty})\) is used.

The third group of axioms describes the Galois group of the algebraic closure of the field \(F\) (asserts that the Galois group of the closure of the field is \(\hat Z\) in Theorem 3).

In the proof of Theorem 3 there is also a fourth group of axioms, by means of which the finite fields different from \(F_1,F_2,\ldots\) are excluded.

For the constructed systems of axioms it is proved that they are systems of axioms for the theory of the field \(F\) in Theorem 2 and for the theory of the class \(\{F_1,F_2,\ldots\}\) in Theorem 3, whence, from the recursiveness of these systems of axioms (in the case of recursiveness of \(\chi_F\) in Theorem 2 or of the fulfillment of all the hypotheses of Theorem 3), decidability follows. The undecidability of the theory of the field \(F\) in the case of nonrecursiveness of \(\chi_F\) in Theorem 2 is obvious.

Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR

Received
18 VI 1966

REFERENCES

1. Yu. L. Ershov, I. A. Lavrov et al., Uspekhi Mat. Nauk, 20, no. 4, 37 (1965).
2. S. Lang, Introduction to Algebraic Geometry, N. Y., 1958.
3. S. Lang, A. Weil, Am. J. Math., 76, no. 4, 819 (1954).
4. A. Robinson, Théorie métamathématique des idéaux, Paris, 1955.
5. N. Bourbaki, Algebra, Multilinear Algebra, Ordered Groups, Moscow, 1965.