* Disclaimer: may not actually be brief.
Galois Theory is a really elegant area of mathematics. It relates two different topics - abstract algebra and geometry - to produce results that are greater than the sum of the parts.
Galois Theory has possibly the most romantic history of any mathematics ever. It was originated by Evariste Galois, a student in France during the 19th century. He was a firebrand of a revolutionary, and kept getting kicked out of universities as a result. Eventually, it seems, someone in government decided that he was too much trouble to have around.
So they set up a honeypot Before Evariste knew where he was, he had been trapped into a pistol duel over a woman (whose identity we don't know). His opponent was one of the greatest duelists in Paris. Galois knew he didn't stand a chance.
The last night of his life, he spent scribbling down diverse thoughts about mathematics. In the margins, he scrawled comments like "I don't have the time! I don't have the time!". The following morning, he went out to fight, and was shot down and left to die. His notes were passed on to a maths professor.
After languishing in various inboxes, the papers reached someone who could see them for the genius they were, and Galois became one of the great heroes of mathematics. It turned out that he'd cracked a set of problems that had been around since the ancient Greeks.
Geometry is the study of shapes - triangles, hexagons, spheres, etc. This subject naturally gives rise to a number of interesting questions, one of which involves the concept of constructibility. A shape is constructible if it can be drawn using only an unmarked ruler ("straightedge") and a compass.
For example, the triangle can be constructed by the following procedure:
Et voila, one beautiful equilateral triangle. So a three-sided shape is constructible.
A four-sided shape turns out to be very easy. It turns out to be possible to bisect any angle*, so just take a straight line (technically an angle of 180°) and bisect it. That gives you two lines at 90° to each other - a right angle. Draw two more perpendicular lines and you've got yourself a square.
Well, a rectangle actually. But the idea's the important thing :)
So you can draw a triangle and you can draw a square. Can you draw a pentagon? The answer turns out to be "yes", but the procedure is so intricate and icky that I'm not even going to try to explain it. By this time, mathematicians were coming to realise that a new approach was needed.
That's where abstract algebra came in. To be continued...
* Procedure left as an exercise for the interested reader. Or google it. What do I care?