After the development of the Cubic and Quartic Formulas during the early to mid 1500s, mathematicians all over Europe worked to discover a formula for solving the general quintic equation, an equation of the form x^{5}+bx^{4}+cx^{3}+dx^{2}+ex+f=0. For 250 years they failed.

During the early 1800s, two mathematicians began their brief careers and each would prove, separately, that a general equation of the 5th degree was not solvable using the four standard mathematical operations and extraction of roots.

Niels Abel was born in Norway in 1802. Around 1824, he proved that the general quintic equation is not solvable. He had his proof printed and sent copies to many of the leading mathematicians in Europe who, because Abel was not very well known at the time, ignored it.

Abel died in 1829 as a result of poverty and ill health. In 1830, the French mathematician Cauchy found a copy of Abel’s paper and eventually published it in 1841. Abel was also awarded the Grand Prix of the French Academy for his work in 1830.

Evarist Galois was born in France in 1811. In 1830, he subbmitted a paper for consideration for the Grand Prix (which was eventually awarded to Abel). His paper was taken home by one of the judges (Jospeh Fourier) who died shortly thereafter. Galois’ paper was lost and not considered for the prize.

After his death in 1832, Galois’ brother and one of his friends collected Galois’ papers and delivered them to the mathematician Liouville who worked over them for for the next ten years. Finally, by the 1840s, the work of both Abel and Galois was recognized for the genius it was.

Galois’ proof in particular used the idea of permutations, or the number of ways that objects can be combined as the basis for his proof. This concept of permutation laid the foundation for the development what became known as Group Theory and Abstract Algebra throughout the 18-1900s.

If you’ve never heard of group theory, it can be one of the most accessible and fun areas of mathematics. It also has important applications in physics and computer science.

As I mentioned in a previous post, there is a book called *The Equation That Couldn’t Be Solved* by Mario Livio that outlines the history of these ideas.

Have a great holiday break!