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=8
^{2}+3^{2}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*2*.^{2n}+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

*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

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