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The differential equation $x^{-1}dx = dy$ then [i.e. assuming $K\setminus \lbrace 0\rbrace$ simply connected] engenders a global isomorphism between the multiplicative group $(x\neq 0$) and the additive group $(y)$.

 

(Original: "Die Differentialgleichung $x^{-1}dx=dy$ vermittelt daher eine globale Isomorphie zwischen der multiplikativen Gruppe $(x\neq 0)$ und der additiven Gruppe $(y)$.")

Question 1: How to understand the highlighted sentence? In particular, is my interpretation correct and if yes, how exactly to show such an $F$ is a group isomorphism $K^\times \simeq (K,+)$?

 

Question 2: Does this prove Gelfand-Mazur, as Ebbinghaus seems to imply, or merely Ostrowski's theorem, as Witt's own title claims? If it only proves Ostrowski's theorem, is there a way to upgrade this to a full proof of Gelfand-Mazur?

The differential equation $x^{-1}dx = dy$ then [i.e. assuming $K\setminus \lbrace 0\rbrace$ simply connected] engenders a global isomorphism between the multiplicative group $(x\neq 0$) and the additive group $(y)$.

(Original: "Die Differentialgleichung $x^{-1}dx=dy$ vermittelt daher eine globale Isomorphie zwischen der multiplikativen Gruppe $(x\neq 0)$ und der additiven Gruppe $(y)$.")

Question 1: How to understand the highlighted sentence? In particular, is my interpretation correct and if yes, how exactly to show such an $F$ is a group isomorphism $K^\times \simeq (K,+)$?

Question 2: Does this prove Gelfand-Mazur, as Ebbinghaus seems to imply, or merely Ostrowski's theorem, as Witt's own title claims? If it only proves Ostrowski's theorem, is there a way to upgrade this to a full proof of Gelfand-Mazur?

I suspect this double page would contain some hint or useful background information; annoyingly, this free online copy of Hille stops at page 265, and all other copies I have found are later editions with significantly changed chapters. Can someone find these two pages of Hille's original book?

which Dan Petersen scanned (thanks!), see comments. Unfortunately, I do not see any clue in there, it just seems to refer back to Lorch and Mazur and the "Liouville-style" proof.

Background: As sort of a hobby, Ernst Witt gave extremely short proofs for famous theorems. This question is about his six-line proof of the Gelfand-Mazur Theorem ("The only $\Bbb C$-Banach algebra $K$ which is a skew field is $\Bbb C$ itself") resp. its historic predecessor and now corollary, Ostrowski's Theorem ("The only complete Archimedean fields are $\Bbb R$ and $\Bbb C$"; not to be confused with the more famous Ostrowski's theorem which classifies valuations on $\Bbb Q$). Witt's article is "Über einen Satz von Ostrowski", Arch. Math. 3 (1952), p. 334, reprinted on p. 404 of his Collected Papers. Unfortunately, I do not have access to either source right now. The best I could find is this (H.-D. Ebbinghaus, Numbers, p. 245) English translation of its decisive three sentences.

I understand the first sentence, which says that w.l.o.g. we can assume $\dim_{\Bbb R}K>2$ and hence $K^\times$ simply connected. I also understand the third sentence which says that there cannot be an isomorphism between the additive group of $K$ and the multiplicative group $K^\times$ (obviously, as we are in characteristic $0$; there is a typo in the translation, since of course it's the element $-1$ which is of order 2 in $K^\times$, and IIRC that's what Witt writes in the original). But the second sentence

Note 4: Witt's paper has four references, two of which are Ostrowski's and Mazur's papers with the repective theorems, the two others are

E. Hille, Functional Analysis and Semi-Groups (Amer. Math. Soc. Coll. Publ. XXXI). New York 1948, pp. 474–475, and

E. R. Lorch, The theory of analytic functions in normal abelian vector rings. Trans. Amer. Math. Soc. 54, pp. 414–425 (1943). (Thanks Daniele Tampieri for the link.)

I suspect they would contain some hint or useful background information; I do not have access to the first source. The second one quotes Mazur for the theorem (Thms 3/4 p. 417) but does define the logarithm as a path integral starting from the unit element (p. 422).

Background: As sort of a hobby, Ernst Witt gave extremely short proofs for famous theorems. This question is about his six-line proof of the Gelfand-Mazur Theorem ("The only $\Bbb C$-Banach algebra $K$ which is a skew field is $\Bbb C$ itself") resp. its historic predecessor and now corollary, Ostrowski's Theorem ("The only complete Archimedean fields are $\Bbb R$ and $\Bbb C$"; not to be confused with the more famous Ostrowski's theorem which classifies valuations on $\Bbb Q$). Witt's article is "Über einen Satz von Ostrowski", Arch. Math. 3 (1952), p. 334, reprinted on p. 404 of his Collected Papers. The best free online source I could find is this (H.-D. Ebbinghaus, Numbers, p. 245) English translation of its decisive three sentences.

I understand the first sentence, which says that w.l.o.g. we can assume $\dim_{\Bbb R}K>2$ and hence $K^\times$ simply connected. I also understand the third sentence which says that there cannot be an isomorphism between the additive group of $K$ and the multiplicative group $K^\times$ (obviously, as we are in characteristic $0$; there is a typo in the translation, since of course it's the element $-1$ which is of order 2 in $K^\times$, and that's what Witt writes in the original). But the second sentence

Note 4: Witt's paper has four references, two of which are Ostrowski's and Mazur's papers with the respective theorems. He also refers to

E. R. Lorch, The theory of analytic functions in normal abelian vector rings. Trans. Amer. Math. Soc. 54, pp. 414–425 (1943) (thanks Daniele Tampieri for the link)

but only by saying that it presents a different proof via imitating complex function theory over $K(i)$ (as the function $f(z) = (z-a)^{-1}$ for $a \in K \setminus \Bbb C$ would be a bounded entire function and thus contradict Liouville's theorem). Incidentally, Lorch's paper quotes Mazur (Thms 3/4 p. 417) but also does define the logarithm as a path integral starting from the unit element (p. 422).

The last remaining reference is

E. Hille, Functional Analysis and Semi-Groups (Amer. Math. Soc. Coll. Publ. XXXI). New York 1948, pp. 474–475

I suspect this double page would contain some hint or useful background information; annoyingly, this free online copy of Hille stops at page 265, and all other copies I have found are later editions with significantly changed chapters. Can someone find these two pages of Hille's original book?