UDC 519.48

MATHEMATICS

Yu. L. ERSHOV

ON THE ELEMENTARY THEORY OF MAXIMAL NORMED FIELDS

(Presented by Academician A. I. Mal'tsev on 9 VI 1965)

1. We shall consider objects of the form \(\langle F, v, \Gamma\rangle\), where \(F\) is a field; \(v\) is a (non-Archimedean) norming of the field \(F\); \(\Gamma\) is the value group of the norming \(v\). We shall call these objects, not quite precisely, normings. \(F_v\) will denote the residue field of the norming \(v\). The notation \(\langle F_1, v_1, \Gamma_1\rangle \subseteq \langle F_2, v_2, \Gamma_2\rangle\) will mean that \(F_1 \subseteq F_2\), \(\Gamma_1 \subseteq \Gamma_2\), \(v_1(a) = v_2(a)\) for \(a \in F_1\); in this case the norming \(v_2\) is called an extension of the norming \(v_1\). If \(n = [F_2 : F_1]\) is finite, then \(e = [\Gamma_2 : \Gamma_1]\) is the ramification index of the norming and \(f = [(F_2)_{v_2} : F_{1v_1}]\) is the relative degree of the norming, finite, and \(n \ge f \cdot e\) \((^2)\).

If \(\langle F_1, v_1, \Gamma_1\rangle \subseteq \langle F_2, v_2, \Gamma_2\rangle\) and \(F_{2v_2} = F_{1v_1}\), \(\Gamma_2 = \Gamma_1\), then \(v_2\) is called an immediate extension of \(v_1\). If a norming has no proper immediate extensions, then it is called maximal. Examples of maximal normings are provided by fields of formal power series \(\langle F(t^\Gamma), v, \Gamma\rangle\) \((^3)\) and local fields \((^4)\).

We shall call a norming \(\langle F, v, \Gamma\rangle\) algebraically complete if: 1) for every finite algebraic extension \(F_1\) of the field \(F\) there exists a unique (up to the naturally defined equivalence of normings over \(\langle F, v, \Gamma\rangle\)) norming of the field \(F_1\) such that \(\langle F, v, \Gamma\rangle \subseteq \langle F_1, v_1, \Gamma_1\rangle\); 2) for every finite algebraic extension \(F_1\) of the field \(F\) and norming \(\langle F_1, v_1, \Gamma_1\rangle\) satisfying condition 1), the equality \(n = f \cdot e\) holds, where \(n = [F_1 : F]\), \(e\) is the ramification index, \(f\) is the relative degree.

Lemma. If \(\langle F, v, \Gamma\rangle\) is a maximal norming, then it is algebraically complete.

Let \(\langle F, v, \Gamma\rangle\) be a norming; a well-ordered sequence \(\{a_\rho\}\) of elements of \(F\), having no last element, is called pseudo-convergent if \(v(a_\rho - a_\sigma) > v(a_\sigma - a_\tau)\) for \(\rho > \sigma > \tau\). An element \(a \in F\) is called a limit of the pseudo-convergent sequence \(\{a_\rho\}\) if \(v(a - a_\rho) = v(a_{\rho+1} - a_\rho)\) \((^3)\). A norming \(\langle F, v, \Gamma\rangle\) is called \(\xi\)-complete, where \(\xi\) is an ordinal number, if every pseudo-convergent sequence \(\{a_\rho\}_{\rho<\beta\le\xi}\) of elements of \(F\) has a limit in \(F\) \((^3)\). If \(\xi\) is the least ordinal of cardinality \(\aleph_\alpha\), then instead of \(\xi\)-completeness we shall speak of \(\aleph_\alpha\)-completeness.

1. We shall regard normings \(\langle F, v, \Gamma\rangle\) as models of the signature

\[ \sigma = \langle F^1, \Gamma^1, V^2, Q^2, S'^3, S^3, P^3\rangle, \]

where the predicate \(F^1\) singles out the elements of \(F\), \(\Gamma^1\) the elements of \(\Gamma\); \(V^2(x,y)\) means that \(x \in F\), \(y \in \Gamma\), and \(v(x) = y\); \(Q^2\) is the order relation \(\le\) on \(\Gamma\); \(S'^3\) is the addition predicate in \(\Gamma\); \(S^3, P^3\) are the predicates of addition and multiplication in the field \(F\).

Let \(T_0\) be a system of first-order sentences of the signature \(\sigma\), whose models are precisely all algebraically complete normings. \(T_0\) can be chosen recursively \((^1)\).

If \(\mathfrak A\) is an arbitrary formula, then \(\mathfrak A^\Gamma\) will denote the formula obtained from \(\mathfrak A\) by relativizing the quantifiers with respect to the predicate \(\Gamma^1\). If \(T\) is a set of formulas, then \(T^\Gamma = \{\mathfrak A^\Gamma \mid \mathfrak A \in T\}\). If \(\mathfrak A\) is arbitrary—

free formula, then \(\mathfrak A^{F_v}\) will denote the formula obtained from \(\mathfrak A\) by relativizing the quantifiers with respect to the formula

\[ A_v(x)=V^2(x,y)\ \&\ Q^2(0,y)\ (v(x)\ge 0) \]

and replacing \(x=y\) by

\[ (\exists z)S^3(y,z,x)\ \&\ (\exists t)(V^2(z,t)\ \&\ Q^2(0,t)\ \&\ (t\ne 0))\quad (v(x-y)>0). \]

If \(T\) is a set of formulas, then

\[ T^{F_v}=\{\mathfrak A^{F_v}\mid \mathfrak A\in T\}. \]

1. Theorem 1. Let \(\langle F,v,\Gamma\rangle\) and \(\langle F_1,v_1,\Gamma_1\rangle\) be two algebraically complete normings, \(F_v\simeq F_{1v_1}\) fields of characteristic \(0\), \(\Gamma\simeq \Gamma_1\), \(\operatorname{Ext}^1(A,F_v^*)=0\) for every torsion-free abelian group \(A\). If \(\overline F=\overline{F_1}=\aleph_{\alpha+1}\) and \(\langle F,v,\Gamma\rangle,\langle F_1,v_1,\Gamma_1\rangle\) are \(\aleph_\alpha\)-complete, then \(\langle F,v,\Gamma\rangle\) is isomorphic to \(\langle F_1,v_1,\Gamma_1\rangle\).

Theorem 2. Let \(\langle F,v,\Gamma\rangle\) and \(\langle F_1,v_1,\Gamma_1\rangle\) be two algebraically complete normings; \(F_v\) a field of characteristic \(0\). \(\langle F,v,\Gamma\rangle\) is arithmetically equivalent to \(\langle F_1,v_1,\Gamma_1\rangle\) if and only if \(F_v\) is arithmetically equivalent to \(F_{1v_1}\), and \(\Gamma\) is arithmetically equivalent to \(\Gamma_1\).

The proof of Theorems 1 and 2 uses the results of the papers \((^{1,3,4})\), and the idea of the proof of these theorems is analogous to the idea of the proof of the corresponding theorems in \((^3)\); in particular, the proof uses the technique of ultraproducts.

Theorem 3. Let \(\mathfrak F\) be a class of fields of characteristic \(0\); \(\mathfrak G\) a class of ordered abelian groups;

\[ \mathfrak F(\mathfrak G)=\{\langle F,v,\Gamma\rangle\mid \langle F,v,\Gamma\rangle \]

is a maximal norming, \(F_v\in\mathfrak F\), \(\Gamma\in\mathfrak G\}\). If \(T_1\) is a system of axioms of the theory of the class \(\mathfrak F\); \(T_2\) is a system of axioms of the theory of the class \(\mathfrak G\), then

\[ T_0\cup T_1^{F_v}\cup T_2^\Gamma \]

is a system of axioms of the class \(\mathfrak F(\mathfrak G)\).

Theorem 4. Let \(\mathfrak F\) be a class of fields of characteristic \(0\), \(\mathfrak G\) a class of ordered abelian groups. If the theory of the class \(\mathfrak F\) is decidable, and the theory of the class \(\mathfrak G\) is decidable, then the theory of the class \(\mathfrak F(\mathfrak G)\) is also decidable.

1. Let \(F\) be a field of characteristic \(p\ne 0\). We shall say that \(F\) satisfies assumption A \((^4)\) if, for every equation of the form

\[ x^{p^n}+a_1x^{p^{n-1}}+\cdots+a_nx=b,\qquad a_i,\ b\in F, \]

there is at least one root in \(F\).

Theorem \(1'\). Let \(\langle F,v,\Gamma\rangle\) and \(\langle F_1,v_1,\Gamma_1\rangle\) be two algebraically complete normings; \(F_v\simeq F_{1v_1}\); \(F,F_1\) fields of characteristic \(p\ne 0\); \(F_v\) satisfies assumption A; \(\Gamma\simeq\Gamma_1\); \(p\Gamma=\Gamma\); \(\operatorname{Ext}^1(A,F_v^*)=0\) for every torsion-free abelian group \(A\). If \(\overline{F_1}=\overline F=\aleph_{\alpha+1}\) and \(\langle F,v,\Gamma\rangle,\langle F_1,v_1,\Gamma_1\rangle\) are \(\aleph_\alpha\)-complete, then \(\langle F,v,\Gamma\rangle\) is isomorphic to \(\langle F_1,v_1,\Gamma_1\rangle\).

Theorem \(2'\). Let \(\langle F,v,\Gamma\rangle\) and \(\langle F_1,v_1,\Gamma_1\rangle\) be two algebraically complete normings; \(F,F_1,F_v\) fields of characteristic \(p\ne 0\); \(F_v\) satisfies assumption A; \(p\Gamma=\Gamma\). \(\langle F,v,\Gamma\rangle\) is arithmetically equivalent to \(\langle F_1,v_1,\Gamma_1\rangle\) if and only if \(F_v\) is arithmetically equivalent to \(F_{1v_1}\), and \(\Gamma\) is arithmetically equivalent to \(\Gamma_1\).

Theorem \(3'\). Let \(\mathfrak F\) be a class of fields of characteristic \(p\ne 0\) satisfying assumption A; \(\mathfrak G\) a class of ordered abelian groups \(\Gamma\) such that \(p\Gamma=\Gamma\);

\[ \mathfrak F(\mathfrak G)=\{\langle F,v,\Gamma\rangle\mid \langle F,v,\Gamma\rangle \]

is a maximal norming, \(F\) is a field of characteristic \(p\), \(F_v\in\mathfrak F\), \(\Gamma\in\mathfrak G\}\). If \(T_1\) is a system of axioms of the theory of the class \(\mathfrak F\); \(T_2\) is a system of axioms of the class \(\mathfrak G\), then

\[ T_0\cup T_1^{F_v}\cup T_2^\Gamma\cup\{(\forall x)(px=0)\} \]

is a system of axioms of the theory of the class \(\mathfrak F(\mathfrak G)\).

Theorem \(4'\). Let \(\mathfrak F\) be a class of fields of characteristic \(p\ne 0\) satisfying assumption A; \(\mathfrak G\) a class of ordered abelian groups \(\Gamma\) such that \(p\Gamma=\Gamma\). If the theory of the class \(\mathfrak F\) is decidable, and the theory of the class \(\mathfrak G\) is decidable, then the theory of the class \(\mathfrak F(\mathfrak G)\) is decidable.

1. All ordered abelian groups \(\Gamma\) considered in this section will satisfy the following restriction:

There exists a least positive element in \(\Gamma\).

\[ (*) \]

We shall denote this element by \(l\).

We shall say that the norming \(\langle F, v, \Gamma\rangle\) is absolutely unramified if, in the case when \(F_v\) has characteristic \(p\ne 0\),

\[ v(p)=v(\underbrace{1+\cdots+1}_{p\ \text{times}})=I. \]

Theorem \(2''\). Let \(\langle F,v,\Gamma\rangle\) and \(\langle F_1,v_1,\Gamma_1\rangle\) be two algebraically complete absolutely unramified normings. \(\langle F,v,\Gamma\rangle\) is arithmetically equivalent to \(\langle F_1,v_1,\Gamma_1\rangle\) if and only if \(F_v\) is arithmetically equivalent to \(F_{1v_1}\), and \(\Gamma\) is arithmetically equivalent to \(\Gamma_1\).

Let \(T_0'=\{\mathfrak A_p\mid p\) is a prime number; \(\mathfrak A_p\) is the formula meaning that if \(v(p)>0\), then \(v(p)=I\}\).

Theorem \(3''\). Let \(\mathfrak F\) be a class of fields; \(\mathfrak G\) a class of ordered abelian groups satisfying condition \((*)\); \(\mathfrak F(\mathfrak G)=\{\langle F,v,\Gamma\rangle\mid \langle F,v,\Gamma\rangle\) is an absolutely unramified maximal norming; \(F\in\mathfrak F,\ \Gamma\in\mathfrak G\}\). If \(T_1\) is a system of axioms for the theory of the class \(\mathfrak F\); \(T_2\) is a system of axioms for the theory of the class \(\mathfrak G\), then \(T_0\cup T_0'\cup T_1^F\cup T_2^\Gamma\) is a system of axioms for the theory of the class \(\mathfrak F(\mathfrak G)\).

Theorem \(4''\). Let \(\mathfrak F\) be a class of fields, \(\mathfrak G\) a class of ordered abelian groups satisfying condition \((*)\). If the theory of the class \(\mathfrak F\) is decidable and the theory of the class \(\mathfrak G\) is decidable, then the theory of the class \(\mathfrak F(\mathfrak G)\) is decidable.

1. All results on the decidability of elementary theories of fields obtained in \((^1,^3)\) are consequences of the theorems indicated above. For example, from Theorem \(4''\) it follows that, for any prime number \(p\), the theory of the field \(Q_p\) of \(p\)-adic numbers is decidable.

Theorems 4, \(4'\), \(4''\) make it possible to indicate many different fields and classes of fields with decidable theory; for example, an infinite field of characteristic \(p\ne 0\) with decidable theory that is not algebraically closed. The class of fields of formal power series \(\{F, F\{t_1\},\ldots,F\{t_1\},\ldots,\ldots\{t_n\},\ldots\}\) has a decidable theory if the field \(F\) of characteristic 0 has a decidable theory, or the class of fields of formal power series \(\{F(t^\Gamma)\mid \Gamma\in\mathfrak G\}\), if \(F\) is a field of characteristic 0 with decidable theory, \(\mathfrak G\) is an arbitrary class of ordered abelian groups with decidable theory, etc.

Received
5 VI 1965

CITED LITERATURE

\(^1\) Yu. L. Ershov, Algebra and Logic, seminar, 4, no. 2, 1965, p. 5.
\(^2\) O. Zariski, P. Samuel, Commutative Algebra, 2, Moscow, 1963.
\(^3\) J. Ax, S. Kochen, Diophantine Problems over Local Fields, I, II, III, Preprint, Ithaca, N. Y., 1964.
\(^4\) I. Kaplansky, Duke Math. J., 9, 303 (1942).
\(^5\) J.-P. Serre, Corps locaux, Paris, 1962.