1

Proposition 3. If $K$ is an ordered field, then $K$ has a subfield isomorphic to $\mathbb{Q}$.

Exercise 2. Prove Proposition 3 by showing that $φ:\mathbb{N}→K$ defined by $φ(n)=1+...+1$, with $1+...+1$ being $1$ added $n$ times, extends to an embedding of $\mathbb{Q}$ in $K$.

How to solve Exercise 2?

How does embedding, I suppose in meaning of embedding defined before in the post linked as order-preserving ring homomorphism, $e:\mathbb{Q}\,{\rightarrow}\,K$, imply that for all ordered fields $K$ there exists a subfield of $K$ isomorphic to $\mathbb{Q}$?

Community
  • 1
asdasdf
  • 769

2 Answers2

3

If $e:\mathbb{Q}\rightarrow K$ is an embedding, then the image of $e$ is a subfield of $K$ which is isomorphic to $\mathbb{Q}$.

For solving exercise 2, how do you think the extension of $\varphi$ should be extended? What, for instance, should $-{5\over 3}=-{1+1+1+1+1\over 1+1+1}$ be sent to?

(Incidentally, when thinking about these problems, it may be best to explicitly distinguish between the natural number $1$ and the multiplicative identity $1_K$ of the field $K$. So, $\varphi(1+1+...+1+1)=1_K+1_K+...+1_K+1_K$.)

Noah Schweber
  • 245,398
2

First extend $\varphi:\mathbb N\to K$ to $\psi:\mathbb Z\to K$ by $$\psi(n)\ =\ \begin{cases} \varphi(n) & \text{if} & n>0 \\\\ 0 &\text{if}& n=0 \\\\ \left(-1_K\right)\varphi(-n) &\text{if}& n<0 \end{cases}$$ Now extend $\psi:\mathbb Z\to K$ to $e:\mathbb Q\to K$ as follows: for $r\in\mathbb Q$, write $r=\dfrac mn$ where $m,n\in\mathbb Z$, $n>0$, and either $m=0$ or $\gcd(m,n)=1$; then define $$e(r)\ =\ \psi(m)\left[\varphi(n)\right]^{-1}$$ There will be a few things you need to check, such as that $\varphi(n)\ne0_K$, that $e$ preserves ordering, and so on.

George Law
  • 4,103