# Mathematics 351, Abstract Algebra Applications to Classical Problems

Abstract algebra was developed to solve modern problems, but it also solved some very old ones, which had been considered for over 2000 years.

Here is a brief indication of some outstanding examples. For fuller information, you can consult a history of mathematics site such as the one at St. Andrew's College.

Number theory
• In any fraction of the form 1/p with p a prime, the length of the periodic part is a divisor of p-1.
• Example: 1/13=076923 076923 076923 ...: 6 is a divisor of 12.
Reference: See Fermat's Little Theorem- however, you will need my explanation in class as to how this is relevant.
• An odd prime number is the sum of two squares if and only if its remainder, on division by 4, is 1.
• Example: 73=82+32 and 73=4.18+1; but 21=4.5+1 is not a sum of two squares (and not prime, either!)
Reference: Fermat.

Euclidean geometry
• Angle trisection: It is impossible to trisect an arbitrary angle using ruler and compass alone.
• Example: While some angles can be trisected, the 60° angle is a good example of one that cannot be.
This is one of three Classical Greek Problems (the first three listed at that site) which were shown to be unsolvable using modern algebra.
Reference: Fermat. "Trisecting an Angle".
• A regular polygon with p sides (where p is prime) cannot be constructed by ruler and compass unless p has the form 22n+1.
• Example: The first few numbers of this form are
n=1: p=3
n=2: p=17
n=3: p=257
n=4: p=65,537
So for p a prime less than 100,000, the polygon cannot be constructed unless p is one of these four numbers. The next number on the list (for n=5) would be 4294967297, but this is equal to 641.6700417, so this is not prime and the theorem does not apply (this polygon is not constructible, in fact).
Reference: this discovery is due to Gauss.

Solutions to equations
• There is no explicit formula for the solution to the general polynomial equation of degree 5 (or any higher degree).
• Background: The quadratic formula has been known for 4000 years. Explicit solutions to equations of degree 3 or 4 are one of the earliest results of European mathematics (17th century).
This result led to the development of group theory.
Reference: Abel and Ruffini, Galois

Main page, 351.