Cyclic algebra

Remarks

It is the crossed product algebra for the factor set $f(\sigma^i, \sigma^j) = 1$ if $i + j < n$ and $f(\sigma^i, \sigma^j) = a$ if $i + j \ge n$, for $0 \le i, j < n$. So it is a central simple algebra over $K$ of degree $n$, split by $L$. It is isomorphic to $M_n(K)$ if and only if $a$ is a norm from $L$ to $K$, that is, $a = N_{L/K}(b)$ for some $b \in L$.

Example: $L = \C$, $K = \R$, $\sigma$ complex conjugation and $a = -1$ give $\mathbb{H}$. Since $N_{\C/\R}(z) = \abs{z}^2 \ge 0$, the number $-1$ is not a norm, and $\mathbb{H}$ is not split, as expected. With $a = 1 = N_{\C/\R}(1)$ one gets $M_2(\R)$.

Non-example: if $L/K$ is Galois but not cyclic, such as $\Q(\sqrt{2}, \sqrt{3})/\Q$, there is no single generator $\sigma$, and the construction does not apply.

If the characteristic of $K$ is not $2$, the quaternion algebra $\left(\frac{a, b}{K}\right)$ is the cyclic algebra $(K(\sqrt{a})/K, \sigma, b)$ when $a$ is not a square in $K$.