How does Schur determine Galois groups of truncated exponential series?

Question

On Serge Lang (page.274 Example 8) and several algebra textbooks, it goes as $$f_n=\sum_{m=0}^n \frac{X^m}{m!}\qquad \operatorname{Gal}(f)=\begin{cases} \mathfrak{S}_n& 4\nmid n\\ \mathfrak{A}_n& 4\mid n\end{cases}$$ It is claimed that the result is due to Schur, but the reference is in German and the full text is unavailable. Are there any textbook introducing it, or some review on it? And I know nothing about $f_n$, what can we say about it, such as its discriminant?

Answer 1

I can calculate the discriminant of $f_n$. It may be more useful to calculate $F_n=n! f(n)$, which is a monic polynomial of integer coefficients. $$\Delta=\frac{(-1)^{\frac{n(n-1)}{2}}}{1}\operatorname{Res}(F_n,F'_n)= (-1)^{\frac{n(n-1)}{2}}\det\left(\begin{matrix} 1 & n &\ldots & \ldots &n!\\ & \ddots & \ddots & \ddots & \ddots & \ddots\\ &&1 & n &\ldots & \ldots & n!\\ n& \ldots & \ldots & n!\\ & n& \ldots & \ldots & n! \\ && \ddots & \ddots & \ddots & \ddots \\ &&&n& \ldots & \ldots & n!\\ \end{matrix}\right)=(-1)^{\frac{n(n-1)}{2}}\det\left(\begin{matrix} 1 & &\\ & \ddots \\ &&1 \\ n& \ldots & \ldots & n!\\ & n& \ldots & \ldots & n! \\ && \ddots & \ddots & \ddots & \ddots \\ &&&n& \ldots & \ldots & n!\\ \end{matrix}\right)=(-1)^{\frac{n(n-1)}{2}}(n!)^n$$ It explains why when $4|n$, one has $\operatorname{Gal}(f)\subseteq \mathfrak{A}_n$, and $4\nmid n$ not.

A text book says that the result is derived by module $p$, this means for any prime $p\leq n$, $f_n$ always has multiple root (which is clear by direct observation), and has no multiple root for primes $p>n$. If the idea works, this means that for some prime $p>n$, $f(x)$ splits into ``severe'' factors, and one can use it to get enough information of $\operatorname{Gal}(f)$, which I think impossible with routine method of module $p$.

Answer 2

There is a nice (and short) proof of this result using Newton polygons in "On the Galois groups of the exponential Taylor polynomials", by Robert F Coleman.

A good exposition of the topic may be found here: https://mattbaker.blog/2014/05/02/newton-polygons-and-galois-groups/

Answer 3

You might find my diploma thesis from 2010 useful which looks at exactly this question and shows how Schur and Coleman did it. It is written in German though. :-)