# nt.number theory – Explicit construction of division algebras of degree 3 over Q

In his book Introduction to arithmetic groups, Dave Witte Morris implicitly gives a construction of central division algebras of degree 3 over $$mathbb{Q}$$ in Proposition 6.7.4. More precisely, let $$L/mathbb{Q}$$ be a cubic Galois extension and $$sigma$$ a generator of its Galois group.If $$p in mathbb{Z}^+$$ and $$p neq tsigma(t)sigma^2(t)$$ for all $$t in L$$, then
$$D=left{ begin{pmatrix} x & y & z\ psigma(z) & sigma(x) & sigma(y)\ psigma^2(y) & psigma^2(z) & sigma^2(x) end{pmatrix} :(x,y,z)in L^3 right}$$
is a division algebra.

On page 145, just before Proposition 6.8.8, Morris claims that it is knows that every division algebra of degree 3 arises in this manner. This should follow from the fact that every central division algebra of degree 3 is cyclic. I could not find this explicit construction in my references (e.g. Pierce – Associative Algebras, though maybe I missed something) and I would like to know if there is a reference or a quick way to see that this exhausts all central division algebras of degree 3 over $$mathbb{Q}$$.