# Abel and Galois (and Arnold)

Question Is there a connection between Abel and Galois theories of polynomial equations?

Recall that for every polynomial $p(x)\in \mathbb{Q}[x]$ (say, without the free coefficient), Abel considered the monodromy group of the Riemann surface of the analytic function $w(z)$ defined by $p(w(z))+z=0$. There is an expression of $w(z)$ in radicals if the monodromy group is solvable (is it an "if and only if" statement?).

On the other hand, for every polynomial $p(x)\in \mathbb{Q}[x]$, Galois considered the automorphism group (the Galois group) of the splitting field of $p$. The roots of $p(x)$ are expressed in radicals if and only if the Galois group is solvable.

The question asks whether there are any known connections between the monodromy group of $w(z)$ and the Galois group of $p(x)+z$ (considered as polynomial in $x$).

I am pretty sure this is well known. I just cannot find it in the literature.

Update 1. What I called "Abel's proof" of Abels' theorem is in fact Arnold's proof written by Alexeev (an English translation can be found here). Abel's proof was based on different ideas, see this text. So some instances of the word "Abel" above should be replaced by "Arnold". Added "Arnold" to the title.

Update 2. I found a very nice book by Askold Khovanskii, "Topological Galois Theory", where Arnold's proof and its strong generalizations to other types of equations, including differential equations, are explained in detail. Highly recommended.

• Does this mean that Abel discovered the concept of Riemann surfaces before Riemann ? Cause Abel died in 1829, when Riemann was only 3 years old. – Sylvain JULIEN Mar 22 '18 at 17:33
• @SylvainJULIEN It is more of a surprise that Fermat had a proof of his theorem maybe around 1670, when Wiles had approximately $-283$ years of age. – Shalom Mar 22 '18 at 17:42
• @SylvainJULIEN As I recall, according to Pesic's book to "Abel's proof", one of the things which made Abel hard to read was that he worked with the language of multivalued analytic functions, which was later cleaned up by introducing the language of Riemann surfaces. Abel certainly understood that $\sqrt{z}$ is $2$-fold branched at $0$, but (as I understand it) he wouldn't have thought of that as a map from some other surface to $\mathbb{C}$. – David E Speyer Mar 22 '18 at 18:22
• @SylvainJULIEN: More amazingly, Galois discovered his theory about 60 years before abstract groups were defined. Most people explaining Galois theory now follow Artin's proof. – Mark Sapir Mar 22 '18 at 18:59
• I sometimes think Galois would have been able to prove RH had he lived long enough. The fact that a teenager could do what he did is somewhat scary. – Sylvain JULIEN Mar 22 '18 at 19:07

## 2 Answers

The action of the monodromy group of $w(z)$ on the fiber $p^{-1}(a)$ for a non-critical value $a$ of $p$ (that is $|p^{-1}(a)|=\deg p$) is the same as the action of the Galois group of $p(x)+z$ over $\mathbb C(z)$ on the roots of $p(x)+z$ in some splitting field. One can see this by comparing each of these groups with the deck transformation group of the cover $X\to\mathbb P^1\mathbb C$, where $X$ is the normal hull of the cover $P^1\mathbb C\to P^1\mathbb C$, $x\mapsto p(x)$. (Remark: Though it doesn't make a difference, it is slightly more convenient to work with $p(x)-z$ instead of $p(x)+z$.)

• So the monodromy group of $w(z)$ and the Galois group of $p(x)-z$ over $\mathbb{C}(z)$ are isomorphic? Can you give a reference to it? – Mark Sapir Mar 22 '18 at 20:45
• @MarkSapir: It seems to me that Section I of a paper by Joe Harris ( projecteuclid.org/euclid.dmj/1077313717) covers that. Also, one finds the ingredients in Forster's Lectures on Riemann Surfaces, one essential piece being Theorem 8.12. – Peter Mueller Mar 22 '18 at 21:56

Yes, this was explored at length in the work Joseph Fels Ritt in the 1920s-1930s, who wrote it all up rather well, but it also appears in various Serre books (Topics in Galois Theory, I am pretty sure).

I am sure it appears in Serre's books, and certainly Topics in Galois Theory talks a lot of the function field case. As for Ritt, here is a random example (he wrote a lot on this in the twenties, most quite relevant):

Ritt, J. F., On algebraic functions which can be expressed in terms of radicals., American M. S. Trans. 24, 21-30 (1923). ZBL49.0717.01.

Another good source from all this is Jean-Pierre Tignol's book:

Tignol, Jean-Pierre, Galois’ theory of algebraic equations, Singapore: World Scientific. xiii, 333 p. (2001). ZBL0972.12001.

• Are you sure about Serre's book? Could you provide a reference to Ritt's work? – Mark Sapir Mar 22 '18 at 17:19
• Ritt's paper does not contain the word "Galois". Do you know an answer to my question? – Mark Sapir Mar 22 '18 at 18:53
• And Tígnol's book does not contain the word "monodromy". – Mark Sapir Mar 22 '18 at 19:07