PURE MATHEM ATICS 4331
PM 4331: Galois Theory
Scipio del Ferro is believed to have solved the cubic equation x3 + px = q but it was not until 1545 that
Girolamo Cardano published Ars Magna, which contained the solution of Niccolo Fontana (nicknamed
Tartaglia). Cardano also published in Ars Magna a method, due to Ludovico Ferrari, of solving the quartic
equation by reducing it to the cubic.
All the formulae discovered had one striking property, which can be illustrated by Fontana’s solution of x3
+ px = q.
The expression is built up from the coefficients p and q by repeated addition, subtraction, multiplication,
division, and extraction of roots. Such expressions became known as radical expressions. Since all
equations of degree less than five were now solved, it was natural to ask how the quintic equation could
be solved by radicals.
It was not until 1824 that Abel proved conclusively that the general quintic equation could not be solved
by radicals. Galois proved in 1832 that the general polynomial equation of degree five or higher could not
be solved by radicals. The core of this course is a proof of this fact.
Text. The book Field Theory and its Classical Problems by Charles Robert Hadlock, The Carus
Mathematical Monographs #19 MAA, matches the course in content and level. Other references include
Abstract Algebra by Larry Joel Goldstein, Prentice-Hall, and Galois Theory by Ian Stewart, London,
Chapman and Hall Mathematics Series.
Marks. Usually 50% of the final grade is determined by assignments and midterm test and 50% from a
Calendar description. Irreducible polynomials and field extensions. Galois groups and the solution of
equations by radicals.
Prerequisite: M 2051 and PM 3320.
Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, A1C 5S7