By Peter Roquette (Heidelberg)

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Extra resources for History of Valuation Theory

Example text

For any a ∈ K × , there are only finitely many primes p of K such that a p = 1. II. Every prime of K is either archimedean or discrete with finite residue field. 57 Then, if K is of characteristic 0, it follows that K is an algebraic number field of finite degree. If K is of characteristic p > 0 then K is a finitely generated algebraic function field of one variable over a finite field – if it is assumed that K admits at least one prime to avoid trivialities. Today these fields are called “global fields”.

Herbrand had given in [1931b] a method how to change the enumeration of the ramification groups such as to become coherent with the projection of G onto its factor groups H. Today we would describe it by the Hasse-Herbrand function t dx ϕ(t) = 0 (V0 : Vx ) where Vx is defined for real x ≥ −1 to be Vi when i is the smallest integer ≥ x. ) This function ϕ(t) is piecewise linear, monotonous and 71 The existence of such O would today be proved by Zorn’s Lemma; at the time of DeurL ing’s paper he used the equivalent theorem that every set can be well ordered.

Schmidt’s construction is somewhat elaborate and not easy to check. Indeed, it seems that nobody (including his co-author Hasse) did check the details at the time, and his proof was generally accepted. The first who really worked on the proof and checked the details seems to have been Mac Lane. In [1939] he pointed out an error in the proof. (This error concerned both the equal and the unequal characteristic case). Mac Lane politely speaks of an “unproven lemma”, and he gives a new proof of the Structure Theorem (in both cases), not depending on that lemma.