= Simple field extension
{title2=$F(\alpha)$}
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 $[F(\alpha):F]=\deg m_\alpha$. The classes of $1,t,\ldots,t^{d-1}$ give a basis when $d=\deg m_\alpha$.
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.
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