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
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
- 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!)
- 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.
"Trisecting an Angle".
- A regular polygon with p sides (where p is prime)
cannot be constructed by ruler and compass unless p has the
- Example: The first few numbers of this form are
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).
- n=1: p=3
- n=2: p=17
- n=3: p=257
- n=4: p=65,537
- Reference: this discovery is due to
- 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.
Abel and Ruffini, Galois
Main page, 351.