How to solve the 4th degree equation

How to solve the 4th degree equation

Johan S. R. Nielsen No Comment
Bachelor Projects

Keywords: number theory, groebner bases, galois theory

Prerequisites: 01018: Discrete Mathematics 2

In high-school you learned how to solve a 2nd degree equation with a closed expression: if \(ax^2 + bx + c = 0\) then \(x = \frac {-b}{2a} \pm \frac {\sqrt{d}}{2a}\), where \(d = b^2 – 4ac\). You might even have learned how to derive that equation. What happens if we start with a 3rd degree equation, 4th degree or even higher? Mathematicians spent hundreds of years searching for similar closed expressions, and finally succeeded, after enormous amounts of experimentation and clever tricks, in finding such closed expressions for the 3rd and 4th degree. The 5th degree — the quintic — resisted all attacks, and it was finally proved by Abel in 1823 that such an expression is impossible to obtain!

This project is about deriving the expression for the 2nd, 3rd and 4th degree equations in a systematic manner. This is possible using a pinch of Galois theory and a huge helping of Gröbner basis computation. The project involves using a computer algebra system for performing the computations.