The Impossible Quintic Made as Simple as Possible


David Abraham Kault – College of Science Technnology and Engineering, James Cook University, Townsville, Queensland, Australia
Graeme Sneddon – College of Science and Engineering, James Cook University, Townsville, Queensland, Australia
Sam Kault – University of Queensland, Brisbane, Queensland, Australia

Series: Theoretical and Applied Mathematics
BISAC: MAT002000; MAT002010; MAT030000

“This book presents a simple and readable account of Galois’ proof almost 200 years ago that almost all degree five equations cannot be solved by any standard method. The prologue presents an attractive and engaging introduction to the aim and promise of the book. This book provides access to this wonderful mathematics for anyone with a mathematics training up to the level of university entrance. In summary, this is a fascinating book, full of exciting new concepts for a reader coming straight from school. For such readers it would be a challenging read. The book traces the variety of mathematics involved in exploring an amazing proof of a stunning mathematical truth.” – Cheryl E. Praeger, AC, FAA, Emeritus Professor of Mathematics at the University of Western Australia; former president of the Australian Mathematical Society and a recipient of many distinguished international awards

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.

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Table of Contents


1. Introduction to Galois’ Proof
1.1 Radicals and symmetry
1.2 Book outline

2. Theorems related to symmetry
2.1 Vandermonde Determinants
2.2 Symmetric expressions
2.3 Resolvents

3. Euclid’s algorithm and shared roots
3.1 Euclid’s algorithm
3.2 Roots shared with an irreducible polynomial equation

4. Galois’ Proof, Stage 1
4.1 The Resolvent Equation
4.2 Introducing the Galois Group (First Stage of Galois’ Proof, Part 1)
4.3 The Galois set of arrangements (First Stage of Galois’ Proof, Part 2)
4.4 Permutations, presentations and groups
4.4.1 Permutations and permutation notations
4.4.2 Permutation composition
4.4.3 Presentations of a group
4.4.4 Column permutations
4.5 Groups and Subgroups, Modern and Galois
4.6 Presentations and subgroups
4.7 Normal subgroups
4.8 Proof that the Galois’ set defines a group (First Stage of Galois’ Proof, Part 3)
4.8.1 Galois’ Proposition 1
4.9 Automorphisms
4.10 Continuing the proof that the Galois’ set defines a group (First Stage of Galois’ Proof, Part 4)

5. Galois’ Proof, Stage 2
5.1 Fields of numbers
5.2 Roots of unity
5.3 Expanding the field of numbers using radicals (Second Stage of Galois’ Proof, Part 1)
5.4 Using alternative roots of unity
5.5 Adjoining radicals implies normal subgroups Second Stage of Galois’ Proof, Part 2
5.6 Indirect Permutation Specification
5.7 Completing Galois’ Proposition 2 (Second Stage of Galois’ Proof, Part 3)

6. Final steps to the impossible quintic
6.1 Listing the steps
6.2 Proof the Galois group of irreducible quintics contains a 5-cycle
6.2.1 Transitivity
6.2.2 Transitivity implies a 5 cycle
6.3 Proof that some Galois groups associated with irreducible quintics contain a 2-cycle: Permutations of Complex Conjugacy
6.4 Group generated by a 5 cycle and a 2 cycle
6.5 The group of all permutations of 5 objects S5, does not have a normal subgroup of index 5
6.6 Specifying an irreducible quintic with exactly two complex roots
6.7 Factorisation over rationals
6.8 An insoluble quintic

7. Summary


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