Low Order Polynomials, An Introduction

Introduction

You are given a polynomial of degree 4 or less, with coefficients taken from an integral domain. If the domain is not a field, embed it in its fraction field. For instance, the polynomial may have integer coefficients, but we'll work with rational numbers, and roots of rational numbers.

Closed formulas exist for the roots of these polynomials. By "closed formula", we mean a prescribed set of operations on the coefficients, consisting of addition, subtraction, multiplication, division, and taking n^th roots. For centuries, the greatest mathematicians tried to find closed formulas for the solutions to higher degree polynomials. In 1830 Galois proved there are no such formulas. The solutions to a quintic polynomial, degree 5, cannot be expressed algebraically. You can approximate them numerically, but there is no formula. To my mind this is one of the most beautiful theorems in mathematics, but we can't present the proof here. You need a solid background in field extensions and galois groups.

Galois is one of the most brilliant mathematicians who has ever lived. It's a pity he didn't live longer. He died at age 20, in a duel. apparently his marksmanship was not equal to his mathematical skill. fortunately for us, he wrote down most of his ideas prior to his death; thus spawning two new branches of mathematics, group theory and galois theory. Read more about this amazing, troubled man.