5
$\begingroup$

When I was studying class field theory, I saw local class field theory. However, I suddenly became curious about local fields, not local class field theory. As far as I know, the local field is the output result of the localization of global field. So, I would like to know the historical reason why the local field became necessary for some reason.

$\endgroup$
1
  • $\begingroup$ "the local field is the output result of the localization of global field" Be careful. The term localization usually refers to an algebraic process: localizing a commutative ring at a multiplicative subsets (e.g., the complement of a prime ideal). Local fields arise from number fields by a process of completion rather than that algebraic method of localization. $\endgroup$
    – KCd
    Commented Dec 28, 2023 at 2:34

1 Answer 1

9
$\begingroup$

In hindsight, Archimedean local fields (real and complex numbers) were around since 16th century, but nobody thought of them in such terms. What stimulated the development of the abstract concept was the first example of non-Archimedean local field, that of $p$-adic numbers. The motivation was to find an analog of Laurent expansions of functions for number theory. Although Hensel introduced $p$-adic numbers explicitly only in 1897, related constructions appeared earlier in works of Kummer, Dedekind and Weber, see Perrin, A journey throughout the history of $p$-adic numbers and Gouvea, Hensel's p-adic Numbers: early history:

"Hensel’s Idea. To study behavior of a function “near $\alpha$”, use Taylor or Laurent series in powers of $(X - \alpha)$ to supplement the algebraic approach of Dedekind and Weber. To study an algebraic number, use an expansion in terms of powers of a prime number $p$... At first, Hensel talked of finding the $p$-adic expansion of a given algebraic number. Then, he started considering the set of all possible expansions as an object of independent interest. At first, basically a formal construction. Later, various attempts to justify the “convergence” of $p$-adic series."

It was Hensel who initiated the ideology that led to the development of valuations and local fields, and it is to study $p$-adic numbers that Kürschák introduced the concept of valuation (of rank 1 only) in 1912. Hasse then extensively applied it to number theory after 1923, as Hensel originally intended. It was also probably under Hasse's influence that the term "local" got attached to fields in 1930s, before him "small/large" was in circulation, see Math SE, Why the terminology "global fields" and "local fields". Other major early contributors include Ostrowski and Krasner.

General valuations of arbitrary rank were introduced by Krull in 1932, who was building also on the earlier work of Hahn (1907) and Baer (1927) on ordered fields, see Roquette, History of Valuation Theory:

"Although the formal defnition of valuation had been given by Kürschák [1912] it will appear that the ideas which governed valuation theory in its first (pre-Krull) phase all came from Hensel. Thus Hensel may be called the father of valuation theory. Well, perhaps better "grandfather" because he never cared about the formal theory of valuations but only for his $p$-adic number fields... It was Hasse who successfully introduced and applied valuation theoretic ideas into number theory. He always propagated that the valuation theoretic point of view can be of help to better understand the arithmetic structure of number fields. Van der Waerden, a witness of the times of the 1920s and 1930s, speaks of Hasse as "Hensel's best and great propagandist of $p$-adic methods"."

$\endgroup$
1
  • 2
    $\begingroup$ Yes, Hasse resurrected Hensel's idea. The original idea was a bit off-target, but had a good spirit... The technical inaccuricies "put off" some of his contemporaries. And, indeed, even now, thinking that p-adic integers are (best) represented as "power series in $p$" has encouraged-but-hobbled many students. "It turned out" in the early 20th century that many things about finite extensions of $\mathbb Q$, and the corresponding integers, were determined by the "local" analogues. If that had not turned out to be the case, we wouldn't be talking about it. :) Etc. $\endgroup$ Commented Aug 30, 2023 at 22:38

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.