 # ⓘ Galois theory is a subject in mathematics that is centered around the connection between two mathematical structures, fields and groups. Fields are sets of numb .. ## ⓘ Galois theory

Galois theory is a subject in mathematics that is centered around the connection between two mathematical structures, fields and groups. Fields are sets of numbers that have a way of adding, subtracting, multiplying, and dividing. Groups are like fields, but with only one operation often called addition. Fields are often hard to study because there are so many possible combinations of numbers, that the internal structure is hard to decipher. Groups are usually much less complex and easier to understand. Galois theory gives a concrete connection between hard to study fields and easy to study groups.

## 1. Disclaimer

## 2. Historical Motivation

Fields and Groups

The solution to these problems arrived out of the study of fields and groups. It turns out that each equation has a special field and group associated with it.

### 2.1. Historical Motivation Euclid and Geometry

In the year 300 BCE, the Greek mathematician Euclid pronounded yook-lid wrote a book on geometry titled Elements. The text contained thirteen books, all on how to solve problems in geometry. Euclid was able to find ways to bisect any angle; that is, he was able to find an angle that was half of any other angle. Euclid was able to do much more, too. He could construct with a straightedge and compass shapes like circles, triangles, squares, pentagons, and others. However, he was unable to find a way to make every possible polygon. For example, Euclid was unable to make a regular 9-gon, a shape with 9 equal sides, using only ruler and compass. Mathematicians, like Euclid, tried to solve other problems as well. If they could cut any angle in two halves, then was there a way to cut any angle into three equal angles? Similarly, was there a way to square the circle find a square with area equal to π, or a square with side length equal to π {\displaystyle {\sqrt {\pi }}}, or double the cubefind a cube with twice the volume of another, by constructing a side of the cube with length 2 3 {\displaystyle {\sqrt{2}}}? These questions remained unanswered for over 2000 years.

### 2.2. Historical Motivation Fields and Groups

The solution to these problems arrived out of the study of fields and groups. It turns out that each equation has a special field and group associated with it.

### 2.3. Historical Motivation Fields

The field associated with each equation has complicated structure. However, you can tell a lot about the structure by how much you can rearrange elements by shuffling without changing the structure.

### 2.4. Historical Motivation Groups and S n

All of the ways of shuffling a field can be turned into a group. There are special types of groups, called the Symmetric Group on n elements labeled S n, which are all the ways of shuffling n things. Symmetric groups are well behaved and easy to work with when you only shuffle 1, 2, 3, or 4 things. If you use 5 or more, the structure of the symmetric group becomes too chaotic.

## 3. Galoiss Solution

Up until the early 1800s, mathematicians were able to find some answers for specific cases of these unsolved problems. Unfortunately, no one had given a reason or, proof for any equation. It was French mathematician Evariste Galois 25 October 1811 – 31 May 1832 who was the first to find a solution, explaining why certain equations like 2nd, 3rd, and 4th degree polynomials did have nice solution formulas, but other equations, like 5th degree and above, cannot have a formula. The answer boiled down to reducing the problem of the equationss field to the equations group; Galois proved that the two had a connection.

### 3.1. Galoiss Solution Galois Realization

Galois noticed that the structure of the groups associated with polynomials were really just the symmetric groups in disguise. If the degree of a polynomial was n, then the group of the polynomial was the symmetric group on n elements. Galois saw that the group of the equation shuffled the roots of the polynomial, and the structure of the shuffles could be analyzed instead.

### 3.2. Galoiss Solution Solving Euclids Problems

Euclid had wanted to find a way of cutting any angle in thirds. Galois showed that if a ruler and compass could be used to make a third of an angle, it would have an associated equation of degree 3. He showed that only powers of two could be made, by analyzing the equations group and field. Galois showed that trisecting an arbitrary angle was impossible. Galois theory could also be used to show that it was impossible to square a circle and double a cube.

### 3.3. Galoiss Solution What Polynomials had Formulas?

Galois used his theory to also show that for polynomials of degree 5 and higher, their associated groups were S n. He also showed that if an equation had a formula to solve it, then the group of the equation would have a nice structure and was not chaotic. Since the structure of the symmetric group is chaotic for n ≥ 5 {\displaystyle n\geq 5}, he showed that no general formula could exist for those polynomials.