Existence of division algebras with center $\mathbb{Q}$ of prime degree In First Course in Noncommutative rings of T.Y.Lam (p.210), the author stated that "It is known that for each $n$, there exists a $\mathbb{Q}$-division algebra $A_n$ of dimension $p_n^2$, with $Z(A_n)=\mathbb{Q}$." Here $p_n$ is a prime number and $Z(A_n)$ is the center of $A_n$.
How to construct $A_n$? 
Thanks!

Let $p$ be a prime. Then there is a prime $q\equiv1\pmod p$, and so
the $q$-th cyclotomic field will have a subfield $K$ of degree $p$
over $\Bbb Q$ with cyclic Galois group generated by $\sigma$. There will
be a prime $\ell$ which remains inert in $K$.
One now constructs $A$ as a crossed product of $K$. It is a free left $K$
module $A=K\oplus Ku\oplus Ku^2\cdots \oplus Ku^{p-1}$ where we define
multiplication by $u^p=\ell$ and $u\alpha=\sigma(\alpha)u$ for $\alpha\in K$.
This is a central simple algebra by the theory of crossed product.
Tensoring with $\Bbb Q_\ell$ will give a division algebra over
$\Bbb Q_\ell$, so $A$ must be a division algebra.
