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Galois theory (pronouncedgal-wah) is a subject inmathematics that is centered around the connection betweenfields andgroups.
Fields are sets ofnumbers (sometimes abstractly calledelements) that have a way ofadding,subtracting,multiplying, anddividing. Groups are like fields, but with only one operation, often called addition (subtraction is considered addition withnegative numbers).
Fields are often hard to study because there are so many possible combinations of numbers. This makes their internal structure 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.
The nature of Galois' work is extremely advanced. It is usually considered out of reach for general audiences, or anyone without a strong technical background inabstract algebra.
This article will go into some detail, mostly defining types of equations and groups, and give a brief overview of the main ideas behind Galois theory. This article may not truly belong onSimple English Wikipedia, though it may be referenced by those who have trouble understandingthe mainWikipedia page, and just need a brief discussion on the subject.
Beyond the first paragraph of this page, some mathematical experience may be required. All things considered, anyhigh schooler should be able to follow the main ideas laid forth in this article.
In the year 300BCE,[1] theGreekmathematicianEuclid (pronounded yook-lid) wrote a book ongeometry titledElements. The text contained thirteen books, all on how to solve problems in geometry.
Euclid was able to find ways tobisect anyangle; that is, he was able to find an angle that was half of any other angle. Euclid was able to do much more, too. Using astraightedge and compass, he could construct shapes likecircles,triangles,squares,pentagons, and others. However, he was unable to find a way to make every possiblepolygon. For example, Euclid was unable to make a regular 9-gon, a shape with 9 equal sides, using onlyruler and compass.
Mathematicians like Euclid tried to solve other problems as well, like these:
These questions remained unanswered for over 2000 years.
In apolynomialequation, you can only add and subtract multiples ofpowers of a variable, like. Annth power is avariable multiplied by itselfn times. Our example polynomial has 5th, 3rd, 2nd, and 1st powers ofx. Thedegree of a polynomial is the largest power (exponent) found in the equation (5, in our example).
In school, students learn about a type of polynomial equation called aquadratic equation. This is an equation where the highest power is two - for example,, or. The equation is quadratic because there is a squared term.
If are numbers and, then is a valid quadratic equation. If we know then we can solve this equation using thequadratic formula:By plugging in numerical values for a, b, and c, we can find the value of x.
Mathematicians hoped to find a simple equation they could use to find the solutions forany polynomial equation - using only addition, subtraction, multiplication, division, and two other simple calculations:powers androots. After years of work, mathematicians were able to find equations for polynomials with degree 3, and eventually degree 4. However, no one was ever able to find a formula for the solutions to a polynomial with degree 5 or higher.
The answers to these problems grew out of the study of fields and groups. It turns out that each equation has a special field and group associated with it.
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.
All of the ways of shuffling a field can be turned into a group. There are special types of groups, called theSymmetric Group on n elements labeled Sn, which are all the ways of shufflingn 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.
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 wasFrench mathematicianÉvariste 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 equations's field to the equation's group. Galois proved that the two had a connection.
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 wasn, then the group of the polynomial was the symmetric group onn 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.
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 equation's 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.
Galois used his theory to also show that for polynomials of degree 5 and higher, their associated groups were Sn. 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 groupis chaotic for, he showed that no general formula could exist for those polynomials.
