It's really nice to see this on here! Around 8 years ago I read "The equation that couldn't be solved"[1] which is a very readable and not-mathy book about this topic. I was absolultely fascinated and ended up enrolling on an Open University mathematics degree which I've just completed. If you are interested in this and fancy something light to read this book is a nice distraction.

It was this problem and it's solution that truely opened my eyes up to the enormous power and abstraction of mathematics.

A much, much less readable book about Galois theory (this is really the cornerstone of the (general) quintic being unsolvable by a formula with radicals) is Fearless Symmettry [2]. That is a book I wish was twice the length, it will explain what a matrix is over pages but then do a drive by with Frobenius numbers. It is also let down by extremely poor typesetting on Kindle. However if you can stomach it, it's probably the only "popular" book on Galois theory that I know of. It focuses on Wiles proof of Fermats Last Theorem.

[1] https://www.amazon.co.uk/Equation-That-Couldnt-Solved-Mathem...

[2] https://www.amazon.co.uk/Fearless-Symmetry-Exposing-Patterns...