How do elements in the algebraic closure of $\mathbb{Q}$ look like?

Question

If one asks give examples of polynomial with coefficients in $\mathbb{Q}$ who don't have zeros in $Q$, simple examples given are: $x^2-3,x^3-3$. All of these have roots of form $(n)^{\frac{1}{m} }$. There are also some other expressions involving nested cube roots (or nested nth and mth root together) which one could write down ( I am not sure how to write the general expression for this)

My question is, is there a classification on how the archetypical algebrical numbers can look like? / Are there still other types of numbers added by the closure which do not take this forms?

Answer 1

"Roots of polynomials with integer coefficients" is pretty much the simplest answer there is. Not all algebraic numbers can be written using nested roots (this is what "Galois theory" was invented to study; the equivalent statement is "not all Galois groups are solvable"). See https://en.wikipedia.org/wiki/Abel–Ruffini_theorem for more information.