Fields isomorphic to $\mathbb{Q}$

Question

We know that certain fields has an isomorphic copy of $\mathbb{Q}$ (for example, every ordered field). But, is there an no trivial explicit example of a field that is isomorphic to $\mathbb{Q}$?

Answer 1

You're not gonna find a field that is isomorphic to $\Bbb Q$ but doesn't "look just like" it: if $K$ is a field isomorphic to $\Bbb Q$ and $1_K$ is the identity element, then $\Bbb Z$ uniquely embeds in $K$ by identifying $a\leftrightarrow a\cdot 1_K$. Then the only option for the isomorphism $\Bbb Q\cong K$ is $a/b\leftrightarrow (a\cdot 1_K)/(b\cdot 1_K)$.

Basically the point is that you'll never see a field isomorphic to $\Bbb Q$ which the author doesn't just call $\Bbb Q$ because the identification is so "explicit". If you want some trivial answer, you can take any countable set $X$ and use a bijection $\Bbb Q\to X$ to put a field structure on $X$ that will make $X$ isomorphic to $\Bbb Q$ as a field.