Looking for a field isomorphic to $\Bbb{Q}$

Question

I am looking for a field that is isomorphic to $\Bbb{Q}$. Could someone kindly give an example, or construct one such field?

Answer 1

Suppose $F$ is a field isomorphic to $\mathbb{Q}$ via the map $\phi : \mathbb{Q} \to F.$

We usually denote the multiplicative identity of any field with $F$ by the symbol $1.$ We also write $1+1=2, 1+1+1=3$ etc, and often we write $ab^{-1}$ as $a/b.$ For clarify I'll use a subscript $F$ for elements inside $F.$

Now since $\phi$ is a field homomorphism, $\phi(1) = 1_F.$ Since $\phi(x+y)=\phi(x) + \phi(y),$ we have $\phi(2) = 1_F + 1_F =2_F.$ Similarly, $\phi(n)=n_F,$ so again by the field homomorphism axioms, $\phi(n/m) =n_F m_F^{-1} = n_F/m_F.$

Now since $\phi$ is surjective, every element of $F$ looks like $n_F/m_F,$ with these symbols not satisfying any special relation like $1+1+1=0$ as that would contradict injectivity (this means $F$ has characteristic $0$).

So basically, if we agree to standard notations for all fields: write the multiplicative identities as $1$, $1+1=2,$ $ab^{-1} = a/b,$ then any field isomorphic to $\mathbb{Q}$ must be exactly $\mathbb{Q}$ itself! Any "other" field isomorphic to $\mathbb{Q}$ will simply be $\mathbb{Q}$ rewritten with strange notation.