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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.


It turns out that two classical mathematical subjects, ruler and compass constructions and solving polynomial equations, can be translated into questions about field theory.


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:


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


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 2n+1 can be a prime only if n itself is a power of 2, say 2k. 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, 232+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 Sn for n>4 is not solvable in the sense of group theory.



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