1.8 The subfield generated by a subset

An intersection of subfields of a field is again a field. Let $F$ be a subfield of a field $E$, and let $S$ be a subset of $E$. The intersection of all the subfields of $E$ containing $F$ and $S$ is obviously the smallest subfield of $E$ containing both $F$ and $S$. We call it the subfield of $E$ generated by $F$ and $S$ (or generated over $F$ by $S$), and we denote it $F(S)$. It is the field of fractions of $F[S]$ in $E$ because this is a subfield of $E$ containing $F$ and $S$ and contained in every other such field. When $S=\{\alpha_{1},...,\alpha_{n}\}$, we write $F(\alpha_{1},...,\alpha_{n})$ for $F(S)$. Thus, $F[\alpha_{1},\ldots,\alpha_{n}]$ consists of all elements of $E$ that can be expressed as polynomials in the $\alpha_{i}$ with coefficients in $F$, and $F(\alpha_{1},\ldots,\alpha_{n})$ consists of all elements of $E$ that can be expressed as a quotient of two such polynomials.

Lemma 1.23 shows that $F[S]$ is already a field if it is finite-dimensional over $F$, in which case $F(S)=F[S]$.

An extension $E$ of $F$ is said to be simple if $E=F(\alpha)$ some $\alpha\in E$. For example, $\mathbb{Q}(\pi)$ and $\mathbb{Q}[i]$ are simple extensions of $\mathbb{Q}.$

Let $F$ and $F^{\prime}$ be subfields of a field $E$. The intersection of the subfields of $E$ containing both $F$ and $F^{\prime}$ is obviously the smallest subfield of $E$ containing both $F$ and $F^{\prime}$. We call it the composite of $F$ and $F^{\prime}$ in $E$, and we denote it by $F\cdot F^{\prime}$. It can also be described as the subfield of $E$ generated over $F$ by $F^{\prime}$, or the subfield generated over $F^{\prime}$ by $F$: