Galois theory

Galois theory is an extremely complex subject. It is usally introduced only in the last year of undergraduate studies (in math!). This means that this project will only aim at understanding the subject in some detail.

The project can be framed as a way to understand the form of the roots of polynomials with integer coefficients. In this sense we can “define” the rational numbers as the roots of a linear polynomial (a line).

• The rational numbers are all of them roots of linear polynomials with integer coefficients.

But once we move to polynomials of higher order, actually rational numbers are not enough to describe the roots of these higher degree polynomials. A simple example is given by the polynomial

$$ x^2 - 2 = 0 $$

That has as solution the square root of 2. This is not a rational number, and to actually prove this is the first part of the project.

• The real number \( \sqrt{2} \) cannot be written as the ratio of two integers \( n/m \).

If we continue to more complicated plynomials, we can easily see that the roots will involve square roots, and also higher roots. For example the number \( \sqrt[5]{7} \) can be seen simply as the real solution of \( x^5 -7 = 0 \).

Galois theory deals with the oposite statement. When can the solutions of a polynomial equation be written as a combination of rational numbers and basic operations like \( +, -, /, \sqrt{}, \sqrt[3]{}, \dots \)?. For example, the solutions of the quadratic equation

$$ x^2 + x - 1 = 0 $$

are given by the expression

$$ x= \frac{-1\pm \sqrt{5}}{2},. $$

There exist a similar formula to solve the cubic equation

$$ x^3 + x - 1 = 0,, $$

whose real solution reads

$$ x = \frac{\sqrt[3]{\frac{1}{2}(9+\sqrt{93})}}{\sqrt[3]{9}} - \sqrt[3]{\frac{2}{3(9+\sqrt{93})}}$$

What is tremendously surprising is that there exist no formula that solves the quintic equation

$$ x^5 - x - 1 = 0,. $$

Note that this is not a statement about our present status of knowledge. What Galois theory proves is that the solution to the previous equation cannot be written using a finite number of symbols \( +, -, /, \sqrt{}, \sqrt[3]{}, \dots\).

• Compute, using Newton’s method an approximation to the solution to the previous equation.

In order to complete this project, you need to understand what is an irrational number, and how to prove that \( \sqrt{2} \) is one of them. You also need to at least understand what Galois theory proves (the wikipedia page is excellent). You will also need to understand the difference between the general statement of the Abel-Ruffini theorem, and the results of Galois theory. There are many other on-line resources about Galois theory on-line. An excellent popular book about the subject is The Equation That Couldn’t Be Solved: How Mathematical Genius Discovered the Language of Symmetry.

A very elementary knowledge of what a group is (in mathematics), will also be needed.