= Power basis
{title2=$1,\alpha,\ldots,\alpha^{n-1}$}
If $K(\alpha)/K$ is a simple algebraic <field extension> of degree $n$, the displayed powers form a <basis>. The minimal polynomial of $\alpha$ reduces every higher power to a <linear combination> of them, while a nontrivial linear relation among the first $n$ powers would give a polynomial of degree below the minimal polynomial vanishing at $\alpha$. If $\alpha$ is an <algebraic integer> over the rational numbers, this is a basis of the order $\mathbb Z[\alpha]$ as a free abelian group; it need not be an <integral basis> of the whole <ring of integers>.
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