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Let $F$ be an algebraically closed field of characteristic zero. Is there a unique copy of the field of rational numbers $\mathbb{Q}$ inside $F$ or there are many?

PS: By a copy of $\mathbb{Q}$ I mean a subfield of $F$ which looks like (i.e. isomorphic as a field to) the rational numbers field.

A related question is: is $\mathbb{Q}$ necessarily a subfield of $F$?

Edit: A third question is: is the real numbers field $R \supset \mathbb{Q}$ unique in $F$?

user48900
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  • Suppose $k$ is a subfield isomorphic to $\mathbb{Q}$ but strictly containing $\mathbb{Q}$. Any ring homomorphism $k \to \mathbb{Q}$ sends the copy of $\mathbb{Q}$ inside $k$ to the entirety of $\mathbb{Q}$. Where can we send the irrational elements of $k$? – Elchanan Solomon Dec 18 '13 at 14:18
  • By "many copies of the rationals inside $;F;$", you mean something like $;\Bbb Q\times\Bbb Q\times\ldots;$ , or what? – DonAntonio Dec 18 '13 at 14:23
  • @PrahladVaidyanathan : $\Bbb Q\cdot i$ is not a subfield of $\Bbb C$. – Andrea Mori Dec 18 '13 at 14:25

2 Answers2

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Let $F$ be a field(not necessarily algebraically closed) and $K\subset F$ a subfield. Then the multiplicative neutral elements $1_F$ and $1_K$ must coincide, otherwise $1_F-1_K\neq0$ would be a zero-divisor in $F$.

Now suppose that $K_1$ and $K_2$ are subfields of $F$ both isomorphic to $\Bbb Q$. By what we just said, must have $L=K_1\cap K_2\neq\emptyset$. If $K_1\neq K_2$, the field $L$ would be identified to a proper subfield of $\Bbb Q$, of which there is none. Therefore $K_1=K_2$.

Moreover, given any field $F$ of characteristic $0$ the natural map $$ \phi:\Bbb Z\longrightarrow F,\qquad \phi(n)=n\cdot1_F $$ is injective (this is actually the definition of characteristic $0$), giving an embedding $\Bbb Z\subset F$.

If $n\neq0$ the element $\phi(n)\in F$ has an inverse in $F$, and thus $\phi$ extends to a map $$ \Phi:\Bbb Q\longrightarrow F,\qquad \Phi(\frac mn)=\Phi(m)\Phi(n)^{-1} $$ which is easily seen to be well-defined and a field embedding.

Andrea Mori
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Any field $F$ of characteristic 0 necessarily includes exactly one isomorphic copy of $\mathbb{Q}$ as a subfield; it is the prime field of $F$ (the smallest subfield containing $1_F$).

universalset
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