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A nice property of algebraically closed fields is that the theory that describes them ($ACF$) admits quantifier elimination: any statement can be shown equivalent (in the theory) to another statement without quantifiers.

This is a crucial point in showing that $ACF_p$, the theory of algebraically closed fields of characteristics $p$ (either $0$ or a prime), is complete: every statement of $ACF$ is equivalent to a statement without quantifiers which basically only assert things on the characteristics (e.g. $(1 + 1 + 1 = 0) \land \lnot (1 + 1 = 0))$.

My question is: when, how and why did people start to have interest in those aspects of the theory of fields? (or simply to ask themselves questions like: if I can formulate a theorem related to $\mathbb{C}$ "in algebraic terms", can it be proven using algebraic properties only?)

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A useful sketch of history is given in Alfred Tarski's Elimination Theory for Real Closed Fields by van den Dries. The result was known to Tarski by 1948 (when his Decision Method for Elementary Algebra and Geometry, referenced as [48$^{\text{m}}$] below, came out), and is simpler than quantifier elimination for real closed fields that he focused on, but he did not make much of it, nor even stated it explicitly. In his earlier work (1937), he did not even spell out "real closed fields" despite listing the axioms for them:

"The title of the monograph suggests a change of emphasis from definability questions to completeness, the fact that the truth of any sentence in R can be established purely on the basis of the axioms for real closed fields... Tarski's axioms here are just the familiar ones for ordered fields plus an axiom schema saying that sign-changing polynomials have a zero. Remarkably, he does refer to van der Waerden's section [1937, p. 229ff.] on real closed fields, but does not mention these fields, and only indicates one other model of his axioms besides $\mathbb{R}$, namely the field of real algebraic numbers.

Lowenheim, Skolem and Langford eliminated quantifiers to settle logical questions about newfangled structures like Boolean algebras and other ordered sets, where a parallel to older concerns of classical algebra was not at all evident. This origin may explain why Tarski refers only once briefly to the connection between quantifier elimination and algebraic elimination theory. Also, Tarski never seems to have explicitly mentioned in print that the theory of algebraically closed fields admits quantifier elimination. He obviously knew this result; cf. [48$^{\text{m}}$, p. 54, Note 16] (and p. 373 of Seidenberg [1954]), but only drew attention to its zero-dimensional consequences: decidability, and completeness in each characteristic.

Now, eliminating quantifiers for algebraically closed fields is, modulo familiar logic, just a simple exercise in the Euclidean algorithm for multivariable polynomials over $\mathbb{Z}$, much easier than doing the same thing for real closed fields. Still, this exercise would have been worthwhile in 1948... In the writings of A. Seidenberg [1954], [1956] and A. Robinson [1956a], [1958], [1959a] the kinship of logical quantifier elimination to algebraic elimination theory, specializing parameters, Nullstellensatze and similar ideas becomes more explicit and very fruitful."

Tarski's work on quantifier elimination generally did not attract attention of algebraists, or, indeed, anyone outside of mathematical logic until 1954. Then, Seidenberg, his colleague at Berkeley, published a proof of Hormander's inequality for polynomials based on it, with logical formalism reduced to a minimum. This made it more accessible, but the full "realization" came, perhaps, in 1956, with Abraham Robinson's monograph Complete Theories. Robinson was attracted by the model-completeness consequence of quantifier elimination, which led him to rethinking some classical algebraic results, and to their generalizations, including for algebraically closed valued fields:

"This model-completeness property drew Abraham Robinson's attention, and in Complete Theories he showed that model-completeness of RCF, the elementary theory of real closed fields, follows on general grounds from two facts known since the classical Artin-Schreier paper [1926]: (i) Each ordered integral domain has a real closure. (ii) If $R$ is a real closed field then a simple ordered field extension $R(x)$, with $x$ distinguished, is up to $R$-isomorphism uniquely determined by the set $\{r \in R: r < x\}$. Besides these two algebraic facts only the simplest model-theoretic techniques (compactness, diagrams) are used in Robinson's argument, which runs to no more than two pages.

Robinson went on to treat in a similar way other important structures: algebraically closed valued fields [1956a] and certain kinds of ordered Abelian groups (Robinson and Zakon [1960])... Robinson made Tarski's theorem not only an immediate consequence of Artin-Schreier theory, but also its culmination: Artin's solution [1927] of Hilbert's 17th problem by means of a delicate specialization becomes in Robinson's hands a memorable one-liner that can be applied over and over again..."

And so it was. More generally, applications to $p$-adic fields were developed largely under Robinson's influence:

"Robinson's simple methods efficiently organize and create new algebraic knowledge, but more important is that these methods suggest fruitful analogies. The theory of real closed fields assumes here the role of paradigm. It is no accident that a basic corner of $p$-adic theory was entirely developed by logically trained mathematicians. We refer here to the breakthrough papers by Ax and Kochen [1965a], [1965b], [1966] and Ersov [1965], [1966], [1967] (who were of course also inspired by other ideas from logic, in particular by the then popular ultraproducts)..."

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  • $\begingroup$ Interesting! I suppose the first proof (by Tarski or other logicians) did not give an effective procedure for eliminating quantifiers from a given statement? Would you know when such a procedure was described first? $\endgroup$
    – Weier
    Sep 30, 2023 at 12:19
  • $\begingroup$ @Weier Depends on what you call "effective". Both Tarski (1930s) and Seidenberg (1954) gave elimination algorithms, both mostly intractable. Collins gave a more efficient algorithm based on cylindrical algebraic decomposition in 1973, see Brown's history sketch. $\endgroup$
    – Conifold
    Sep 30, 2023 at 13:06
  • $\begingroup$ I believe cylindrical algebraic decomposition concerns real closed fields, not algebraically closed fields (although I suppose there are some common methods) $\endgroup$
    – Weier
    Sep 30, 2023 at 14:02

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