An article of Brian Conrad "Impossibility theorems for elementary integration" says that a way of proving the following elliptic integral

If $P(x)$ is a monic polynomial of degree $\ge 3$ without repeated roots, then $$\int \frac{dx}{\sqrt{ P(x)}}$$ is not an elementary integral.

After that, Conrad says that this theorem is a consequence of general facts about compact Riemann surfaces, because the formula obtained by the Liouville theorem for elementary integrals is equivalent to the equality for meromorphic $1$-forms

$$\left({dy}/{y}\right)=\sum c_j (dg_j)/(g_j)+dh$$

on the compact Riemann surface associated to $y^2=P(x)$, and for degree of $P>2$ the left side of the formula is a holomorphic $1$-form on $C$, and a non-zero holomorphic 1-form on a compact Riemann surface never admits an expression like the above expression.

I don't know anything about Riemann surfaces, and my question is: where can I find the information to understand this reasoning? I don´t know if this it's very difficult or it's very elementary in the theory of Riemann surfaces. I appreciate any answer. Thanks

  • $\begingroup$ Make some research on elliptic functions and the compact Riemann surface $\mathbb{C}/(\mathbb{Z}+\tau \mathbb{Z})$. $\endgroup$ – reuns Nov 26 '17 at 1:54

Davenport, James, On the integration of algebraic functions. Lecture Notes in Computer Science, 102. Springer-Verlag, Berlin-New York, 1981.

J. Ritt, Integration in finite terms, Columbia UP, 1948.

Hardy, G. H. The integration of functions of a single variable. Reprint of the second edition, 1916. Hafner Publishing Co., New York, 1971.


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