Paul L. Bailey, Ph.D. - Topics Fall 2006

Course Syllabus

Topic 1: Constructibility
Topic 2: Complex Numbers
Topic 3: Integer Arithmetic
Topic 4: Modular Arithmetic
Topic 5: Polynomial Arithmetic
Topic 6: Irreducibility Criteria
Topic 7: Vector Spaces
Topic 8: Field Extensions [Updated 12/04/06]
Topic 9: Examples [Updated 12/06/06]

The Cosine of 72 Degrees

Problem Set 1
Problem Set 2
Problem Set 2 Solutions

The quadratic formula was proven (geometrically) in Euclid's Elements. The cubic equation was first solved and described in the beginning of the sixteenth century by del Ferro, Tartaglia, and Cardano (with much contention regarding the discovery's attribution). The fourth degree was solved by Ferrari, a student of Cardano. The solution of fifth degree polynomial equations evaded mathematicians, and it was suspected that such a solution did not exist. Ruffini gave the first proof of such a result, although it was incomplete. Abel wrote a successful proof that such a formula, in general, could not exist. Galois, without knowing the work of Abel, described explicit conditions under which solutions can or cannot be written for a given polynomial.

The following link describes this in more detail.
Quadratic, Cubic, and Quartic Equations

The following biographies are as dramatic as they are enlightening.
Scipione del Ferro
Nicolo Fontana Tartaglia
Girolamo Cardano
Lodovico Ferrari
Paolo Ruffini
Niels Henrik Abel
Evariste Galois