Union of infinite subfields of the complex plane

Question

Is the union of all subfields $\mathbb{Q}(\sqrt{n})$ for $n \geq 1$ a subfield of $\mathbb{C}$?

The previous section of the question asked me to prove that the union of an infinite number of subfields $K_n$ with $K_n \subseteq K_{n+1}$ is also a subset of $\mathbb{C}$.

I am struggling to make any ground as I'm not sure how to begin answering the question.

Answer 1

The problem here is that the fields $K_n = \Bbb Q(\sqrt n)$ are not totally ordered for the inclusion.

Let $A$ to be the union of the $K_n$, for $n≥1$. Consider $x=\sqrt 2 \in A$ and $y = \sqrt 3 \in A$. If $A$ was a subfield of the complex numbers, then you should have $x+y \in A$. Is this the case?

If it was the case, then $x+y \in K_n = \Bbb Q(\sqrt n)$ for some $n≥1$. Then we could write $\sqrt 2 + \sqrt 3 = a+b\sqrt n$ for some rational numbers $a,b$. Squaring both sides yields $5 + 2 \sqrt 6 = a^2+nb^2+2ab\sqrt n$... can you continue from there?