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#
Math 237 Introduction to Algebra II

Fall,
2010.

*What this course is about*

### This course is an introduction to field theory and Galois theory. It
includes proofs of the impossibilty of solving certain classical
problems such as "trisecting angles" and "solving 5th degree
polynomials".

### In more detail, recall that a field *K* is a
set equipped with addition, subtraction, multiplication and
division. (This was discussed in Math 236. The symbol *K* is
often used to denote a field because many of the early papers on the
subject were in German, where the word is
*Korper*.) The most familiar example of a field is the rational
numbers *Q*. Formally, most of this course is about extensions
of *Q*, that is larger fields which contain it.

*Ruler and compass constructions*

### If you do not know what ruler and compass constructions are, look
at the links on the ruler and compass
constructions page of this website. Here are some things that can
be done with ruler and compass:

### Bisect an angle.

### Divide a line segment into *n* equal parts for any positive
integer *n*.

### Construct an equilateral triangle, a square, a regular hexagon
and a regular pentagon.

### Construct a line segment whose length is the square root of any
positive rational number.

### These were all known in ancient times. Now here are some things
the ancient Greeks wanted to but were unable to do.

### Trisect an angle, i.e., divide it into three equal parts.

*Square the circle.* Construct a circle having the area of a given
square or vice verse.

*Double the cube.* Construct a cube having double the volume
of a given one.

### Construct regular polygons with other numbers of
sizes. (Classical methods were good for polygons with *n* sides,
where *n* can be any power of 2 times 3, 4, 5 or 15.)

### We now know that the first three of these problems *cannot* be
solved, and the reasons for this will be explained in this course. In
1796 (at the age of 19!) Gauss
found a way to construct a regular 17-sided
polygon, thereby making the first progress on the 4th problem in
2000 years.

### He also determined exactly which polygons are constructible and
which are not. He showed that if *p* is a prime that is 1 more
than a power of 2 (such as 3, 5 and 17), then there is a way to
construct a regular polygon with *p* sides. We know for other
reasons (that I will explain) that a number of the form
2^{n}+1 can be a prime only if *n* itself is a
power of 2, say 2^{k}. The first three values of
*k* (0, 1 and 2) give the primes 3, 3 and 17 mentioned above.
The next two give
*p=257* and *p=65537*. These numbers are called Fermat
primes, and no others are known to exist. The next candidate,
2^{32}+1, is known to be divisible by 641.

*Solving polynomial equations*

### You should know about the quadratic
formula for solving a quadratic equation in one variable. It has
been known since anciant times. There is also a cubic
formula for solving a cubic equation in one variable and a quartic
formula for solving a quartic (fourth degree) equation in one
variable. The latter two were discovered before 1600.

### The problem of solving polynomial equations of degree 5 and higher
remained a mystery until the early 1800s, when it was shown by Abel
and Galois
that no such formula exists. (See links on the quintic equation page for more information.)
Their methods of proof involve field theory and group theory in very
interesting ways. In some sense it boils down to the fact that the symmetric
group *S*_{n} for *n>4* is not solvable in the
sense of group theory.

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