I wanted to write an equation solver for rational polynomials in one variable $X$. However, such solutions do not need to be in $\mathbb{Q}$. What I wanted was to display solutions "lossless", i.e. not $1.414$, but $\sqrt 2$. Somehow, I believe that solutions for Polynomials in $\mathbb{Q}[X]$ are always constructible with "$+$", "$\cdot$" and "$\sqrt[n]{}$".
Now, I have a lot of questions:
- Let's call $K:=\mathbb{Q}(i)(\sqrt[n]{2},\sqrt[n]{3}, \sqrt[n]{5}, ...)$ (brackets mean adjunction). Is $K$ algebraically closed? If not, what field do I need to solve polynomials in $\mathbb{Q}[X]$?
- Is it true that $K$ is exactly the set of algebraic numbers?
- Do you know a simple algorithm for constructing the solutions of a polynomial in $\mathbb{Q}[X]$? For maximum degree $4$, there are formulas, but what if the degree is higher? Important: The algorithm shall not approximate, I would like to have radical terms in the end.