Further Reading
A gentle first reader is the Mir tract by Kurosh. Designed for a one-
semester course is the book by MacDuffee (MacD), who treats selected
topics. The text of Dobbs & Hanks (DH), written for prospective teachers,
is an admirable reference. More comprehensive texts are those of Borofsky
(Bo) and Uspensky (Us), w h o cover most of the topics handled in this
book and provide proofs. Older texts which may be consulted are those
of Burnside (Bu) and Dickson (Di). Chapter II of D.J. Struik’s Source
Book contains excerpts from a number of historical papers on the theory of
equations. For a modern abstract approach to polynomials, the experienced
college student can have recourse to Lausch & N6bauer. An overview is
provided by the essay on the theory of algebraic equations by B.N. Delone,
which is Chapter 4 of Volume 1 of the collection edited by Aleksandrov,
Kolmgorov and Lavrent’ev.
A recent and highly recommended book by J.-P. Tignol deals with the
theory of equations from an historical perspective, recapturing the method-
ology of the pioneers in the field. This treats many of the topics of the first
four and sixth chapters, in particular the question of solvability by radicals.
Chapter 1. Solution of cubic and quartic equations: Bo, Ch. 8; Us, Ch.
V and Bu, Ch. VI, Cajori (Hist), p. 133-139, DH, Ch. 3.
Complex numbers: Bo, Ch. 1.
Chapter 2. Horner’s method: Bo, Ch. 7; Bu, Ch. X, where it is used as
a tool in approximately solving equations.
Graphing: Bu, Ch. I; MacD, Ch. 4.
Multiple roots and derivatives: Bu, Ch. VII; MacD, Ch. 3; DH, Sect. 2.3.
Factor and remainder theorem: DH, Sect. 2.1.
Chapter 3. Greatest common divisor, factorization: Bo, Ch. 2; MacD,
Ch. 7; Us, Ch. I.
Factoring and irreducibility: DH, Ch. 4.
Partial fractions: MacD, Ch. 3.
A survey article with many references on the current state of the art
in factoring polynomials is Susan Laudau, Factoring polynomials quickly,
Notices A.M.S. 34 (1987), Issue 253, 3-8.
Chapter 4. The Fundamental Theorem: Di (1914), Ch. V; Bu, Ch. X.
Uspensky (Us, Ch. X) discusses Lagrange’s method for solving cubic and
biquadratic equations, and provides a different insight into Galois theory.