1.7 The subring generated by a subset

An intersection of subrings of a ring is again a ring (this is easy to prove). Let $F$ be a subfield of a field $E$ , and let $S$ be a subset of $E$ . The intersection of all the subrings of $E$ containing $F$ and $S$ is obviously the smallest subring of $E$ containing both $F$ and $S$ . We call it the subring of $E$ generated by $F$ and $S$ (or generated over $F$ by $S$ ), and we denote it by $F[S]$ . When $S=\{\alpha_{1},...,\alpha_{n}\}$ , we write $F[\alpha_{1},...,\alpha_{n}]$ for $F[S]$ . For example, $\mathbb{C}{}=\mathbb{R}{}[\sqrt{-1}]$ .

Lemma 1.21

The ring $F[S]$ consists of the elements of $E$ that can be expressed as finite sums of the form

equation (1) (1)

\sum a_{i_{1}\cdots i_{n}}\alpha_{1}^{i_{1}}\cdots\alpha_{n}^{i_{n}},\quad a_{% i_{1}\cdots i_{n}}\in F,\quad\alpha_{i}\in S,\quad i_{j}\in\mathbb{N}{}.

Proof.

Let $R$ be the set of all such elements. Obviously, $R$ is a subring of $E$ containing $F$ and $S$ and contained in every other such subring. Therefore it equals $F[S]$ . _□

Example 1.22

The ring $\mathbb{Q}[\pi]$ , $\pi=3.14159...$ , consists of the real numbers that can be expressed as a finite sum

a_{0}+a_{1}\pi+a_{2}\pi^{2}+\cdots+a_{n}\pi^{n},\quad a_{i}\in\mathbb{Q}.

The ring $\mathbb{Q}[i]$ consists of the complex numbers of the form $a+bi$ , $a,b\in\mathbb{Q}$ .

Note that the expression of an element in the form (1) will not be unique in general. This is so already in $\mathbb{R}{}[i]$ .

Lemma 1.23

Let $R$ be an integral domain containing a subfield $F$ (as a subring). If $R$ is finite-dimensional when regarded as an $F$ -vector space, then it is a field.

Proof.

Let $\alpha$ be a nonzero element of $R$ — we have to show that $\alpha$ has an inverse in $R$ . The map $x\mapsto\alpha x\colon R\rightarrow R$ is an injective linear map of finite-dimensional $F$ -vector spaces, and is therefore surjective. In particular, there is an element $\beta\in R$ such that $\alpha\beta=1$ . _□

Note that the lemma applies to every subring containing $F$ of a finite extension of $F$ .