Simple field extension ($F(\alpha )$)

A simple field extension is a field extension generated by one element: $F(\alpha )$ is the smallest subfield of an ambient field containing $F$ and $\alpha$. Its elements are rational expressions in $\alpha$ with coefficients in $F$ and nonzero denominators. If $\alpha$ is algebraic, its minimal polynomial $m_{\alpha }$ identifies $F(\alpha )\cong F[t]/(m_{\alpha })$, so $[F(\alpha ):F]=\mathrm{deg}{m}_{\alpha }$. The classes of $1,t,\dots ,t^{d-1}$ give a basis when $d=\mathrm{deg}{m}_{\alpha }$.

In particular, ${\alpha }^2\in F$ but $\alpha \notin F$ implies $[F(\alpha ):F]=2$. Conversely, in characteristic different from two, any degree-two extension is of this form: for $\beta \notin F$, express ${\beta }^2=a+b\beta$ in the basis $1,\beta$, then $\alpha =\beta -b/2$ has ${\alpha }^2=a+b^2/4\in F$ and generates the same extension. The restriction on the characteristic makes division by two possible.

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.