>>10397407What does a surd as a solution to an equation even mean? People take it for granted, but it's actually (initially) a pretty circular definition. A square root of a number is the number that squares to that number. Hence if I claim that the solution of x^2=2 is the square root of 2, then I'm literally just saying that the solution to that equation is the solution to that equation. It's completely tautological, and doesn't bring anything new.
However, rooted numbers are pretty easy to understand and manipulate, and hence they have become somewhat commonplace. They come from polynomial equations of the form x^n - a = 0, but sometimes they come from slightly more complicated expressions, like x^2 + x + 1 = 0.
A natural question to ask is "what polynomial equations can be solved using only surds?", given that they're such "nice" numbers. An equivalent question is "If I start with the field of rationals Q, and I add elements that are solutions to an equation of the form x^n - a = 0?, what other polynomial equations can I solve?". It turns out that it's always the case for polynomial equations of degree 4 and under, and not always the case for degree 5 and higher, and Galois theory tells you exactly when it can happen.
A natural follow-up question is "what other numbers do I need to add to be able to solve any polynomial equation?". Of course, the preferred objective would be to do this "minimally". If I add all solutions, then of course I can add solve all equations. It turns out that for example, in the degree 5 case, you can completely solve every equation by adding surds and solutions to the equation x^5+x+a=0. The latter terms are not as natural as surds, but they're still pretty simple, and are called Bring radicals.
