Unknown

Notes on Explorations^411

Intelligencer 9 (1987), no. 2, 53-60. Also see P. Erdos & R.C. Vaughan,
Bounds for the rth coefficients of cyclotomic polynomials, .I. Land. Math.
Sot. (2) 8 (1974) 393-400; MR 50, # 9835.
E.37. A recent and somewhat advanced approach to Fermat’s Little The-
orem which obtains the result in the general form

xadp(n/d) E 0 (mod n)

din

where p is the Mobius function (Exploration E.34) is found in C.J. Smyth,
A coloring proof of a generalization of Fermat’s Little Theorem, Amer.
Math. Monthly 93 (1986), 469-470.
E.38. The theory of functions of complex variables treats functions which
can be regarded as generalizations of polynomials and rational functions.
These are assumed to possess derivatives (see the note on Exploration
E.25) except possibly at a discrete set of points on a region of the complex
plane. The principal parts and residues of such functions become significant
in the evaluation of integrals; indeed. some definite integrals of real-valued
functions of a real variable can be evaluated by applying a “calculus of
residues” for the determination of a corresponding complex integral. The
understanding of this theory depends on a background of a second college
calculus course.
For a clear treatment, consult George Polya & Gordon Latta, Complex
Variables, (Wiley, 1974). An older reference which will reward careful study
is Konrad Knopp, Theory of Functions, Part I (Dover, 1952).

E.39 & 40. Probably the best elementary account of the treatment of solv-
ability of equations and ruler and compasses constructions is to be found
in Charles R. Hadlock, Field theory and classical problems, (MAA, 1978:
Carus Monograph # 19). Th ese problems are also treated in D.E. Little-
wood, The Skeleton Key of Mathematics, (Hutchinson University Library,
London, 1949, 1957). An excellent historically sensitive account appears
in Jean-Pierre Tignol, Galois’ Theory of Algebraic Equations, (Longman,
1988). A more advanced treatment of Galois theory is contained in Chapter
4 of Nathan Jacobson, Basic Algebra I. 2nd ed. (Freeman, 1985).

E.41. The theorem that the zeros of the derivative of a polynomial are con-
tained within the smallest polygon containing the zeros of the polynomial
is called the Gauss-Lucas Theorem. A thorough treatment can be found in
Morris, Marden, Geometry of Polynomials, (AMS, 1949, 1966). In Chapter
1, the result is interpreted physically and geometrically, while in Chapter
2, it is established and extended.
A section of problems on this result is found in Part III, Chapter 1,
Section 3 of G. Polya & G. SzegG, Problems and Theorems in Analysis,
(Springer-Verlag, 1972). In W.H. Echols, Note of the roots of the derivative