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 $mathbb{Q}$.