Exercise 4.1.7 (Generated Subfield Examples)

Proof. Here we write $ℱ$ the set of subfields of $ℂ$.

If $X\subset ℂ$, write

the subfield of $ℂ$ generated by $X$.

(i)

We show that $$⟨\{7∕8\}⟩=\mathbb{Q}.$$

Here $X=\{7∕8\}$. Since $\mathbb{Q}$ is the prime field of $ℂ$ (because $\mathrm{char}ℂ=0$ : Lemma 2.3.16), $\mathbb{Q}=\underset{F\in ℱ}{\bigcap <!--\; FUNCTION\; APPLICATION\; -->}F$. But every subfield $F\in ℱ$ contains $\mathbb{Q}$, a fortiori contains $X=\{7∕8\}$, so

$$⟨X⟩=\underset{F\in ℱ,X\subset F}{\bigcap }F=\underset{F\in ℱ}{\bigcap }F=\mathbb{Q}.$$

(ii)

We show that $$⟨\{2+3i\}⟩=\mathbb{Q}(i):=\{a+\mathit{bi}\mid a\in \mathbb{Q},b\in \mathbb{Q}\}.$$

Here $X=\{2+3i\}$. Note that, for every subfield $F\in ℱ$, if $X\subset F$, then $z=2+3i\in F$, thus $i=\frac{z-2}{3}\in F$. Since $\mathbb{Q}\subset F$, every $a+\mathit{bi}\in ℂ$ with $a,b\in \mathbb{Q}$ is in $F$, which proves $\mathbb{Q}(i)\subset F$. Conversely, if $\mathbb{Q}(i)\subset F$, then $F$ contains $2+3i$. This shows the following equivalences, for every subfield $F$ of $ℂ$,

$$X\subset F⟺2+3i\in F⟺i\in F⟺\mathbb{Q}(i)\subset F.$$

Therefore the smallest subfield of $ℂ$ which contains $2+3i$ is the smallest subfield of $ℂ$ which contains $\mathbb{Q}(i)$, that is $\mathbb{Q}(i)$, since $\mathbb{Q}(i)$ is a subfield. More formally

$$⟨\{2+3i\}⟩=⟨X⟩=\underset{F\in ℱ,X\subset F}{\bigcap }F=\underset{F\in ℱ,\mathbb{Q}(i)\subset F}{\bigcap }F=\mathbb{Q}(i).$$

(iii)

We show that $$⟨ℝ\cup \{i\}⟩=ℂ.$$

Here $X=ℝ\cup \{i\}$. Every subfield $F$ of $ℂ$ which contains $X$, contains $ℝ$ and $i$, so contains every $z=a+\mathit{bi}$ with $a,b\in ℝ$. This proves $ℂ\subset F$ (so $F=ℂ$). Therefore the smallest subfield of $ℂ$ which contains $X=ℝ\cup \{i\}$ is $ℂ$. That is

$$⟨ℝ\cup \{i\}⟩=\underset{F\in ℱ,ℝ\cup i\subset F}{\bigcap }F=\underset{F\in ℱ,ℂ\subset F}{\bigcap }F=\underset{F\in ℱ,ℂ=F}{\bigcap }F=ℂ.$$