Changes

=== Algebra ===

=== Algebra ===

# How many positions are there for a Rubik's cube.  Brief discussion of group theory, and how a group acts on the Rubik's cube.

# How many positions are there for a Rubik's cube.  Brief discussion of group theory, and how a group acts on the Rubik's cube.

# Bezout's theorem.  Discuss intersection of conic sections, plane cubics, etc.  Two conics intersect at 4 points, in generally.

 

==== Bezout's theorem ====

It's a very basic fact that two lines intersect at exactly one point.  But we can ask a similar question about other shapes.  At how many points do two parabolas intersect?  A circle and a line?  The graphs of two cubics?  A circle and a line may intersect at either 0 points (for example, the unit circle <math>x^2+y^2=0</math> and the line <math>y=2</math>), 1 point (if the line is tangent to the circle), or 2 points.  It's easy to see that we can arrange for two parabolas to intersect at four points: just consider the examples <math>y=x^2-10</math> and <math>x=y^2-10</math>.  Similarly, we can up with two cubics that intersect at nine points.  You might be noticing a pattern here: a parabola is defined as the set of solutions to a polynomial equation in two variables, whose biggest exponent is 2: we say that a parabola has degree 2.  For the same reasons, we say that a line has degree 1, and a cubic as degree 3.  Bezout's theorem then gives a very rough answer to our question: a curve of degree m and a curve of degree n intersect in at most mn points.

 

We have seen that sometimes the number of intersections is less than this, but Bezout's theorem, properly generalized, says that the number of intersections is in fact ''equal'' to mn: in the case of the circle and the line given earlier, there are exactly two solutions as predicted, if we allow <math>x</math> and <math>y</math> to be complex numbers.  Similarly, a point of tangency is in some sense a "double intersection", and should count twice: if we nudged the line just a little bit, the point of intersection would turn into two distinct points. After we make these adjustments (counting complex solutions, and counting points of tangency multiple times), Bezout's theorem provides a simple and elegant answer to our original motivating question: curves of degree m and degree n intersect at exactly mn points!

 

# Is the quintic solvable in radicals?  And what in the world does this problem have to do with group theory?

# Is the quintic solvable in radicals?  And what in the world does this problem have to do with group theory?

# The ancient Greeks were fascinated by geometric constructions with straightedge and compass.  They could bisect angles and take square roots, but they couldn't trisect angles or take cube roots.  Why not?

# The ancient Greeks were fascinated by geometric constructions with straightedge and compass.  They could bisect angles and take square roots, but they couldn't trisect angles or take cube roots.  Why not?