An Introduction to Galois Theory

This article only skims the surface of Galois theory and should probably be accessible to a 17 or 18 year old school student with a strong interest in mathematics.

What's a Group?

Explore the properties of some groups such as: The set of all real numbers excluding -1 together with the operation x*y = xy + x + y. Find the identity and the inverse of the element x.

Groups of Sets

The binary operation * for combining sets is defined as the union of two sets minus their intersection. Prove the set of all subsets of a set S together with the binary operation * forms a group.

An Introduction to Mathematical Structure

Stage: 5

Article by Emma and Gareth McCaughan

When I started to study "algebra" at university, I was surprised to discover that it looked nothing like the "algebra" I had studied at school. Gone were the algebraic expressions and quadratic equations, and in came a whole new set of words and symbols.

But it was still to do with generalising. In school-level algebra, we can generalise results that work for lots of different numbers (such as $(x-1)(x+1)\equiv x^2-1$,or find a formula that generalises a sequence of numbers ($n^{\textrm{th}}$ term $=3n+4$). The algebra studied at university makes connections between more disparate areas of mathematics, such as arithmetic, combinatorics and symmetry. This is very powerful; if we can show that two situations behave in the same way, then if we find something interesting about one situation, there will be an equivalent result in the other situation.

So algebraists look for ways to describe seemingly different situations in the same way. They will tend to describe them in terms of a set of elements, and one or more operations, which are ways of combining elements. This is quite difficult to understand without seeing some examples, so let's explore some:

1) Imagine taking the numbers 0, 1, 2 and 3. These are the elements . We're going to add them, but we'll do this "mod 4"; that just means that we'll write down the remainder when the answer is divided by 4. This is the operation. So, for example, 2 + 3 = 5 = 1 mod 4.

We can build up a table of the answers we get:
|    |    |    |    |    |
|----|----|----|----|----|
| +  |  0 |  1 |  2 |  3 |
| 0  |  0 |  1 |  2 |  3 |
| 1  |  1 |  2 |  3 |  0 |
| 2  |  2 |  3 |  0 |  1 |
| 3  |  3 |  0 |  1 |  2 |

Here are some more sets of 4 elements, each time with an operation. Try to complete each table, then click here to see if you are right. Some involve arithmetic, some involve symmetry, and one involves looking very silly.
2) Take the numbers 2, 4, 6, 8 and multiply them mod 10 (so just write down the last digit).
|    |    |    |    |    |
|----|----|----|----|----|
| x  | 2  | 4  | 6  | 8  |
| 2  |    |    |    |    |
| 4  |    |    |    |    |
| 6  |    |    |    |    |
| 8  |    |    |    |    |

3) Take the numbers 1, 2, 3, 4 and multiply them mod 5.
|    |    |    |    |    |
|----|----|----|----|----|
| x  | 1  | 2  | 3  | 4  |
| 1  |    |    |    |    |
| 2  |    |    |    |    |
| 3  |    |    |    |    |
| 4  |    |    |    |    |

4) Take the numbers 1, 3, 5, 7 and multiply them mod 8.
|    |    |    |    |    |
|----|----|----|----|----|
| x  | 1  | 3  | 5  | 7  |
| 1  |    |    |    |    |
| 3  |    |    |    |    |
| 5  |    |    |    |    |
| 7  |    |    |    |    |

5) Take a square. Our elements this time will be "the rotations of a square". We could leave it as it is(call this I), we could turn it through 90° anticlockwise (call this Q), we could turn it through 180° (call this Q²), or we could turn it through 270° anticlockwise (call this Q³). Our operation this time is just doing one element after another. So Q²Q³ is turning through 180° then 270°, which is the same as turning through 90°, so the answer is Q.
|    |    |    |    |    |
|----|----|----|----|----|
|    | I  | Q  | Q² | Q³ |
| I  |    |    |    |    |
| Q  |    |    |    |    |
| Q² |    |    |    |    |
| Q³ |    |    |    |    |

6) Take a rectangle. Ideally you need a rectangle of clear plastic, with each corner painted a different colour, but paper will do. We're going to be interested in where the colours move to, but not which way up the plastic is. This time our elements will be the symmetries of a rectangle; the ways we could move it so that it is still in the same orientation. We could leave it as it is (I), or we could flip it vertically or horizontally (V, H), or we could rotate it through 180° (R). Again, the operation is just going to be doing one followed by another; to start you off, VH = R.
|    |    |    |    |    |
|----|----|----|----|----|
|    | I  | V  | H  | R  |
| I  |    |    |    |    |
| V  |    |    |    |    |
| H  |    |    |    |    |
| R  |    |    |    |    |

7) Take a T-shirt, one where the front and back are clearly different. In order that you don't get too many strange looks, you might like to try this out in the privacy of your bedroom! If you take the T shirt off and put it on again, there are four things you could do in between. You could leave it as it is (Same), you could turn it back to front (BTF), you could turn it inside-out (I-O), or you could do both of these (Both). These are our four elements, and the operation is just doing one after another.
|      |      |     |     |      |
|------|------|-----|-----|------|
|      | Same | BTF | I-O | Both |
| Same |      |     |     |      |
| BTF  |      |     |     |      |
| I-O  |      |     |     |      |
| Both |      |     |     |      |

Now have a good look at the tables you have completed, and look at the similarities and differences.

All these tables have a number of things in common:
• the only elements in the table are the ones we started with
• they all have one column and one row which shows the elements in the original order
• each element appears exactly once in each row and column
• they are all symmetrical about the "leading diagonal" (top left to bottom right)
Mathematicians call this structure a group. Not all groups have four elements (they could even have an infinite number), but they all have tables which share most of the properties above.

Put more formally, a group is a set of elements and an operation which have the following properties, where a, b, etc are elements, and * is the operation:
• closure; this means that when we combine two elements, we only get elements which are in the group;
• there is an identity element, e , such that for each element a , e *a = a = a *e
• each element a has an inverse, a -1 , such that a *a -1 = e = a -1 *a
• associativity; this means that if we have an expression involving the operation twice, it does not matter which bit is done first: addition is associative as $a+(b+c)\equiv(a+b)+c$ , but subtraction is not, as $a-(b-c) \neq (a-b)-c$
The first three of these properties have been coloured red, green and blue to show how they relate to the corresponding properties we observed for the tables. The tables we constructed also have the associative property, but that can't easily be seen from thetables.

All the tables we constructed were symmetrical about the leading diagonal, but this symmetry is not part of the definition of a group. However groups that have this property are important enough to have a special name: they're called abelian groups after the Norwegian mathematician Niels Abel. (Many scientists and mathematicians have things named after them; the real challenge is to discover or invent something which is used so much that it loses its capital letter. Abel managed this, even though he died when he was only 26 years old!)

Look at the groups above and identify which element is the identity element in each group. You could also identify the inverse of each element. Some elements are self-inverse ; a *a =e .

We've looked at the properties shared by the tables above. Now look again at the tables. How many different tables are there?

You probably thought there were three. However, try filling in these two again:

2) (multiplication mod 10)
|    |    |    |    |    |
|----|----|----|----|----|
| x  | 6  | 8  | 2  | 4  |
| 6  |    |    |    |    |
| 8  |    |    |    |    |
| 2  |    |    |    |    |
| 4  |    |    |    |    |

3) (multiplication mod 5)
|    |    |    |    |    |
|----|----|----|----|----|
| x  | 1  | 3  | 2  | 4  |
| 1  |    |    |    |    |
| 3  |    |    |    |    |
| 2  |    |    |    |    |
| 4  |    |    |    |    |

Rearranging the elements shows you that all of these examples fit one of just two structures.

We have seen that the tables are not reliable as a way of distinguishing; one useful way to start to describe individual groups is to look at the order of the elements in the group. The order of an element is the number of times it needs to be combined with itself to get the identity element. For instance, if we turn te T-shirt inside-out twice, we're back where we started, so the order of this element is 2. If we rotate a square through 90°, we have to do it 4 times, whereas if we rotate it through 180°, the order is just 2. The identity element obviously has order 1, and elements which are self-inverse have order 2.

There are in fact only two different groups of order 4 (consisting of 4 elements).
V, or K4 (Klein-4) is the name given to a group with 4 elements where all elements other than the identity are self-inverse. (Why V? It's the first letter of the German word for "four".)
C4 is the name given to a cyclic group of 4 elements. A cyclic group is one where one (or more) element isof the same order as the group; all the other elements are created by combining this element with itself.
With a bit of thinking, you may be able to convince yourself that there are no other groups of order 4. There are some hints on this at the end of the article.

This article has been looking at just one kind of mathematical structure, the group. In fact, it's only been looking at groups of order 4. This, of course, barely scratches the surface. Groups can have any number of elements; they can even have infinitely many elements. For instance, take the integers as elements and addition as your operation; the result is a group. (Check those properties!)

Now, the integers are interesting because they have more structure than addition alone gives them: you can multiply integers, too. Multiplication of integers isn't a group operation because most elements don't have inverses: 1/2 isn't an integer. The non-zero integers under multiplication form what's called a semigroup , which more or less means "like a group, but without inverses". But semigroups are pretty boring; matters start to get more interesting when you put addition and multiplication together. The result is a structure called a ring , which means something like "some elements, a group operation, and a semigroup operation, where the two operations are related by the distributive law. This is interesting because rings have enough structure to do all kinds of useful things with, but on the other hand they have little enough structure that lots of things in mathematics either are rings or can easily be turned into rings.
There are lots of different sorts of mathematical structure: semigroups, groups, rings, fields, modules, groupoids, vector spaces, and so on and so on. They're all based on the same insight: that when something interesting (like the integers) turns up, you should try to work out what the basic facts about it are that make it interesting, and then look for other things that share those basic facts -that is, other instances of the same structure. Those links give you a flavour; actually understanding all this stuff takes a lot of time at university!

Mathematicians have even gone one step further and asked: What about this whole business of mathematical structures? What's its structure? The answer to that turns out to be a whole new area of mathematics called "category theory". It's not for the faint-hearted!

Coming down from these stratospheric heights of abstraction, there's an awful lot more even to finite groups than we've seen here. For instance: in a certain (rather complicated) way, all finite groups can be built out of building-blocks called the finite simple groups . (It's a bit like the way that all positive integers can be built out of prime numbers.) There are a few infinite families of finite simple groups, where each member of the family is built in basically the same way; and there are just 26 other finite simple groups that don't belong to any of those families. The biggest of those groups is called "the Monster" and it has exactly 808017424794512875886459904961710757005754368000000000 elements. When mathematicians say something's "simple" they don't mean quite the same as normal people do!

Further exploration

Try proving that there are only two different groups of order 4. The best strategy for this is to try and construct a group which is not K4 . It must have an identity element e, and at least one element a which is not self-inverse, and its inverse a -1 . If you try to construct a table, sooner or later you will establish that the only possibility turns out to be C4 .

We've looked at a couple of symmetry groups; one was the rotations of a square, and the other was the symmetries (both reflections and rotations) of a rectangle.
Using the Shuffles resource , you can explore the symmetries of various regular polygons. For example, the set of symmetries of an equilateral triangle, the set of rotations of a regular hexagon and the set of reflections of a regular hexagon all have six elements. Are they groups? (Check the properties.) Are they the same group?

Are there any elements of order 3 in the groups explored in this article? Can you explain? What about groups with different numbers of elements? What orders are the elements in these groups?

You might like to read the earlier NRICH article Small Groups. There are various problems involving groups in the March 2005 issue of the site. In particular, you might like to look at "What's a group?"

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here.
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