Splitting variety for bicyclic algebras

Question

Let $F$ be a field contains a primitive root of unity of order $p$, where $p$ is a prime number. Let $a,b \in F^\times$, then one can look at the cyclic algebra $(a,b)_p \in {_p}Br(F)$ where ${_p}Br(F)$ is the $p$-torsion part of the Brauer group of $F$. It is a well known fact that $(a,b)_p$ splits iff there exist $X \in F[\sqrt[p]{a}]$ such that $N_{F[\sqrt[p]{a}]/F}(X)=b$. This condition can be written as a simple polynomial equation with coefficients in $F$, so the question about $(a,b)_p$ being split is reduced to whether this equation has a solution.

Now for $a,b,c,d \in F^{\times}$ one may define $(a,b)_p \otimes (c,d)_p$ which is a bicyclic algebra over $F$. It is known that there exists a system of polynomial equations which determines whether it splits or not. Is there a description of this system? I would be glad to have a detailed reference about this material.

Thank you!

Answer 1

For any central simple algebra $A$ you can form the (projective) Severi-Brauer variety $SB(A)$ such that $A$ splits if and only if $SB(A)$ has a rational point, that amounts to a system of homogeneous equations that can be written explicitly, if needed. If you prefer affine varieties you can apply Jouanolou's trick then.