Field extensions, generated subrings $F[S]$, generated subfields $F(S)$, and simple extensions

A field extension $K\mathrm{/}F$ is a field $K$ together with a specified field homomorphism $F\to K$ (Field, Field homomorphism and embedding). Since that map is injective, we identify $F$ with its image and write $F\subseteq K$.

For $S\subseteq K$, the subring generated by $F$ and $S$ is $F[S]=\bigcap \{R:R\text{ is a subring of }K\text{ and }F\cup S\subseteq R\},$ and the subfield generated by $F$ and $S$ is $F(S)=\bigcap \{E:E\text{ is a subfield of }K\text{ and }F\cup S\subseteq E\}\mathrm{.}$ These intersections are nonempty because $K$ is among the displayed subrings and subfields, and they are respectively a subring and a subfield (Subring: a subset containing $1_R$ and closed under addition, additive inverses and multiplication, Subfield: a subring of a field closed under inverses of its nonzero elements, and therefore a field with the restricted operations). Equivalently, $F[S]$ and $F(S)$ are the smallest subring and subfield of $K$ containing $F\cup S$. For a singleton, write $F[a]$ and $F(a)$. An extension $K\mathrm{/}F$ is simple if $K=F(a)$ for some $a\in K$.

For completeness, the asserted injectivity is immediate: if $\phi (a)=0$ with $a\mathrm{\ne }0$, then $1=\phi (a^{-1}a)=\phi (a^{-1})\phi (a)=0$, a contradiction.