In 1832, just before his untimely death, twenty year old French mathematical genius, Everiste Galois spent the whole night rewriting the new mathematics he had discovered. It gave an amazing answer to a mathematical problem from antiquity. He could not know it then, but his new mathematics also enabled our modern world through its application to quantum mechanics and coding theory. His new mathematics wasn't easy and Galois' overly brief writing style had rendered a previous draft of his ideas incomprehensible to the top mathematicians of the day. However, he did know that he might not have much time to give this new mathematics to the world. He was right - he was mortally wounded in a duel the next day. It has since been useful to put Galois theory within a framework of more abstract algebraic concepts, but this has made his work accessible only to those with advanced mathematics. This book follows Galois' original approach but avoids his overly brief style. Instead, unlike other books, it makes Galois' amazing mathematical ideas accessible to those with just university entrance level mathematics. Quadratic equations were solved with the help of square roots in ancient times. Equations with an x3 and those with an x4 term were solved 500 years ago with the help of cube roots and fourth roots, though with increasingly difficult formulas. Galois showed that a formula with square roots, cube roots and fourth and fifth roots, cannot be obtained for the quintic - an equation with an x5 term. It is not just that any potential formula would be so long and difficult that it has not yet been discovered, it is absolutely impossible. The proof of this impossibility is long and occupies this whole book, but readers are rewarded by getting to understand something that at first sight may seem impossible, a proof of impossibility. Readers will also be rewarded by getting to fully understand the series of startlingly clever mathematical manipulations of a genius.