Let $k$ be a field and $A$ be a central simple algebra over $k$. It's known that $A$ has a splitting field (i.e. a field $K/k$ such that $A\_K\cong M\_n(K)$ for some $n$) which is finite and Galois.


This allows us to define a *reduced norm* $N:A\to k$ which is given by the determinant $M\_n(K)\to K$ and then descended (via Galois descent) to $k$.


I wonder if we can do the same thing using fpqc descent but avoiding the need for the existence of a finite Galois splitting field, and using only that $\overline{k}$ splits $A$, which is way simpler. (Of course that's not for didactic reasons, since we're exchanging a hard theorem for a harder one. That's just for my curiosity.)



Existence of reduced norms for CSAs using fpqc descent
You can define the reduced norm using fpqc descent. Have a look at chapter III, section 1.2 in Knus' "[Quadratic and Hermitian Forms over Rings](https://doi.org/10.1007/978-3-642-75401-2)", for instance.


