When it doesn't, we yearn for something that will fill the void so that it does. It's like that note you yearn for in a musical piece that the composer seems to avoid. One yearns for a resolution of the tension.

Complex numbers offers that resolution.

> And why would R be "entitled" to an algebraic closure?

It's the birthright of every field.

Why would N be entitled to it? We made up negative numbers and more just to have a closure. You just learn about them at an age when you don't question it yet.

The good news is that Q is not really entitled to a closure either.