The building blocks of Galois theory

Galois theory: what exactly is it? ---------------------------------------

When faced with a polynomial equation, our first instinct is to ask whether solutions exist, which then allows us to factor it. For example, we can check that 2 and ‒3 satisfy *x ² + x ‒ 6 = 0; we can therefore write *x ² + x ‒ 6 = (x ‒ 2)(x + 3). But what can we do when the equation has no roots?

The idea is to define a larger set of numbers containing a new element that satisfies the equation.

Thus, for the equation *x ² + 1 = 0 over the set ℝ of real numbers, we construct the set ℂ of complex numbers (see opposite). But how do we "add" numbers? This is where the notion of a field comes in: a set equipped with addition and multiplication satisfying the usual properties of arithmetic. The most familiar examples are, of course, ℝ and ℂ, as well as ℚ, the set of rational numbers (fractions). But there are many others! Given a field and a polynomial equation with no solution in that field, Galois theory allows us to construct a "larger" field, called an extension, that is "as small as possible" and in which the equation has a root—or, better still, factors completely.

The Galois group -------------------