2. Development of the theory of equations
Hovedinnhold
In elementary school, we learned to solve the simplest of equations, first degree equations such as
2x - 3 = 5.
Later, this was expanded to systems of equations: e.g. two first degree equations with two unknowns
2x + 3y = 8,
3x - 5y = 1.
In secondary school, we learned to solve second degree equations and we learned about polynomials, expressions such as
x3 + 2x2 - 5x + 7,
and about polynomial division. And one has come so far as when the middle ages drew to a close in Europe. But then, around 1530, came two breakthroughs in the Italian Renaissance. One became able to solve the third degree equation
x3 + a1 x2 + a2 x + a3 = 0,
and just afterwards, the fourth degree equation. As a result of this and the flourishing of mathematics in general, there came three innovations that can be said to belong to algebra. Firstly, mathematical symbols began to be used more and more. At this time the symbols +, -, = and √ (the square root sign) were introduced. In the early 17th century, the convention that x, y and z are used to denote the unknowns, while a, b and c are used to denote known quantities also originated. Secondly, with the use of the formula for the roots of the third degree equation, one could not avoid taking the square root of a negative number, even though the answer was a natural number. Thus, for the first time, one had to use an algebraic system, the complex numbers, which have no direct physical interpretation. Thirdly, the solution of the third degree equation inspired further deeds: can one solve fifth degree equations, do all polynomial equations have solutions?
The following centuries saw a rapid development of mathematics. But, despite the intense effort, no solutions could be found for the fifth degree equation. Then, in 1824, Niels Henrik Abel showed that the general fifth degree equation can not be solved with the algebraic operations of addition, subtraction, multiplication, division and root extraction (but many special fifth degree equations, such as x5 - 3 = 0 can still be solved). His goal was now to understand when arbitrary polynomial equations can be solved. Abel's death in 1829 came too early, but a young Frenchman, Evariste Galois, built further upon Abel's insights and had in 1831 developed a clear picture. Central to understanding when we have algebraic solutions is the equation's symmetries. The symmetries of an object, be it an equation, a football, or a Rubik's cube, is an algebraic structure that is called a group.
Read on about groups.