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

Here are the answers to the challenges:

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	8	2	6
4	8	6	4	2
6	2	4	6	8
8	6	2	8	4

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

x	1	2	3	4
1	1	2	3	4
2	2	4	1	3
3	3	1	4	2
4	4	3	2	1

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

x	1	3	5	7
1	1	3	5	7
3	3	1	7	5
5	5	7	1	3
7	7	5	3	1

5) Rotations of a square

	I	Q	Q²	Q³
I	I	Q	Q²	Q³
Q	Q	Q²	Q³	I
Q²	Q²	Q³	I	Q
Q³	Q³	I	Q	Q²

6) Symmetries of a rectangle

	I	V	H	R
I	I	V	H	R
V	V	I	R	H
H	H	R	I	V
R	R	H	V	I

7) The T-shirt!

	Same	BTF	I-O	Both
Same	Same	BTF	I-O	Both
BTF	BTF	Same	Both	I-O
I-O	I-O	Both	Same	BTF
Both	Both	I-O	BTF	Same

Order of elements in a group. Rotations. Groups. Regular polygons. Structures of simple finite groups. Symmetry. Reflections. Modulus arithmetic. Inverses. Subgroups.

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An Introduction to Galois Theory

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