[Previously asked][1] on Math Stackexchange without answers.

**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][2], 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.](https://books.google.ca/books?id=Z53SBwAAQBAJ&pg=PA245&lpg=PA245&dq=witt%27s+proof+ostrowskis+theorem&source=bl&ots=_rK0IZ-SBA&sig=FgputHkH0h2qtn9v7cdnta7cyJQ&hl=de&sa=X&ved=0ahUKEwj7rpmFiJbbAhVE34MKHdxYDqoQ6AEITTAE#v=onepage&q=witt's%20proof%20ostrowskis%20theorem&f=false)

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

> 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 $(y)$.") 

seems to hide some details. My guess would be the idea is that by simply connectedness, path-integrating $f(x) = x^{-1}$ from the startpoint $1$ to any $x\neq 0$ gives a well defined "logarithm" function $F(x)$ which has the property $F(ab) = F(a) + F(b)$ and is bijective. (Which then, as said, I understand gives a contradiction and shows no such $K$ with $\Bbb R$-dimension $>2$ exists.)

> **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?


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**Note 1**: (**Edit**) As checked by J. Wengenroth (thanks), Witt indeed writes $x^{-1}dx = \color{red}{d}y$, whereas Ebbinghaus has $x^{-1}dx=y$. 

**Note 2**: I am aware of the standard calculus proof that $F(x) = \int_1^x \frac{dt}{t}$ satisfies $F(ab) = F(a) +F(b)$ (actually, teaching that in my calculus class last week reminded me of this question), but it seems Witt is assuming a generalisation of this, and maybe more differential calculus, to a possibly infinite dimensional $\Bbb R$ resp. $\Bbb C$-vector space, which is a bit outside of my comfort zone.

**Note 3**: In the comment sections [here][3] and [here][4] Keith Conrad explains how Witt's argument proves the Fundamental Theorem of Algebra, referring to Ebbinghaus as a source. I understand that explanation, but to my (probably poor) understanding it uses Lie-theoretic background and the assumption that $\dim_{\Bbb R} K < \infty$ which both are absent from Witt's reasoning. I interpret Witt as claiming that his argument works in the much more interesting case $\dim_{\Bbb R} K =\infty$. -- Ebbinghaus himself claims that Witt's proof is "of course, the proof mentioned [on p. 242] based on the exponential function". I doubt this "of course". If Witt wanted to prove the theorem with an explicit exponential series, why would he not write that but a differential equation? Also, at the point (p. 242) where Ebbinghaus discusses that proof of Gelfand-Mazur via the exponential, he admits that a proof of the exp-log correspondence "is incidentally somewhat troublesome, because we have to consider a power series whose terms are power series", which reenforces my belief that Witt might have had a slightly different argument in mind.

**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)][5] (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][6] 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?**


  [1]: https://math.stackexchange.com/q/2789677/96384
  [2]: https://link.springer.com/article/10.1007%2FBF01899369
  [3]: https://mathoverflow.net/a/34718/27465
  [4]: https://math.stackexchange.com/a/2307/96384
  [5]: http://www.ams.org/journals/tran/1943-054-03/S0002-9947-1943-0009090-0/
  [6]: https://archive.org/details/functionalanalys017173mbp