= Splitting criterion for a cyclic algebra
{title2=$(L/K,\sigma,a)\text{ splits}\iff a\in N_{L/K}L^\times$}
If $a=N_{L/K}c$, let $L$ act on itself by multiplication and let the extra generator act by $c\sigma$. Its $n$th power is multiplication by $a$, and these operators give the <cyclic algebra> as $\operatorname{End}_K(L)$. Conversely, if the algebra is a <matrix algebra> on an $n$-dimensional <vector space>, its embedded degree-$n$ field $L$ makes that space one-dimensional over $L$. The extra generator is then $c\sigma$ in an $L$-basis, so its $n$th power forces $a=Nc$.
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