Post #4: Abel and Galois

 The quadratic formula is part of the algebra curriculum around the world. The idea is to find the two solutions to the equation

ax² + bx + c = 0, where a ≠ 0. 

 

The reason for the stipulation is that if a = 0, the equation no longer has a squared term and is not a quadratic.

There are formulae for the solution of higher degree polynomials as well, where the solution is equations using the coefficients, the letters that aren't x.

Cubic: ax³ + bx² + cx + d = 0, where a ≠ 0.

There are formulas to find all three roots

Quartic: ax⁴ + bx³ + cx² + dx + e = 0, where a ≠ 0.

There are formulas to find all four roots.

 

Quintic: ax⁵ + bx⁴ + cx³ + dx² + ex + f = 0, where a ≠ 0.

No formula always works. 

People searched for a set of equations using the coefficients a, b, c, d, e and f that would solve any quintic for centuries, and given the caliber of mathematicians who tried, the general consensus in the field was that the task was impossible, and mathematicians set themselves to a new strategy, find the proof of insolubility.

This was also a tough nut to crack, but it was solved independently by two mathematicians. The Frenchman Evariste Galois and the Norwegian Niels Abel.

Abel published first. Galois saw Abel's work and realized it overlapped with some of the work he had already done, but not published. The work he did was more about an overlying method, which we now call Galois theory, and he had a devil of a time trying to get anyone to pay attention.

There are twists and turns in both the lives of these fellows, but I give you this spoiler. Both die young, Abel of an illness and Galois in a duel.

I give the links for the biographies of Abel and Galois from the math history library at the University of St. Andrews. If you have any interest in the lives of famous mathematicians throughout history, the library provides very good, brief biographies. Both stories are remarkably dramatic and very quick reads, and I recommend both of them to anyone who shows interest in this blog.

Tomorrow: The structure of the subgroups of a finite group G.