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

(I have a math degree, so I don't have any issues with C, but this is the kind of question that would have troubled me in high school.)

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.