Why does this ring have rank $k!$?

Please see the following paper by Bhargava and Satriano here, which contains the details of this computation and other related results.

• M. Bhargava and M. Satriano, On a notion of “Galois closure” for extensions of rings, Journal of the European Mathematical Society 16, 1881-1913 (2014).

The aim of the authors is to define for commutative ring extensions a notion similar to the notion of Galois closure of fields. For a commutative ring $B$ and a commutative $B$-algebra $A$ which is locally free of rank $n$, they construct a commutative ring $G(A|B)$ which is called the $S_n$-closure of $A$ over $B$. If $B\subseteq A$ is a field extension of degree $n$ with Galois group $S_n$, it is easily seen that $G(A|B)$ is exactly the Galois closure of $A$ as a field extension of $B$. The authors investigate several properties of the $S_n$-closure, such as for example the functoriality, and the behavior with respect to base change or to general finite products. They also investigate several cases of natural ring extensions, e.g. monogenic or étale ones. Finally, they extend the notion of $S_n$-closure to schemes.