Today I realised that I didn’t remember how to prove that the complex numbers are a topological field w.r.t. to the metric induced by the Euclidean norm $\|(x,y)\|=\sqrt{x^2+y^2}$.
I think the usual proof is boring and yet elementary, but still I have always thought that the fact that the field structure on $\mathbb{C}$ is compatible with the Euclidean norm is some kind of miracle.
I would be tempted to ask here if people share the same feeling as me that the continuity of the field structure on $\mathbb{C}$ w.r.t. the Euclidean norm is something very deep even if its proof is quite basic and unremarkable, or if I am just deluding myself.
But I realise this question is not appropriate for this site.
I will try to make it more acceptable by asking
Is there some conceptual reason, lurking behind the usual proof, that makes things work? Is this related to the structure of the Galois group of $\mathbb{C}$ over $\mathbb{R}$, or maybe by the fact that the Euclidean norm happens to be the square root of the algebraic norm on $\mathbb{C}$?