1.2. THE FUNDAMENTAL THEOREM OF ALGEBRA 3
Bibliography
[B] F. Beukers, A note on the irrationality of ζ(2) and ζ(3), Bull. London Math. Soc. 11
(1979), 268-272.
[MK1] Dragoslav S. Mitrinović and Jovan D. Kečkić, The Cauchy Method of Residues: Theory
and Applications, D. Reidel Publishing Co., 1984.
[MK2] Dragoslav S. Mitrinović and Jovan D. Kečkić, The Cauchy Method of Residues: Theory
and Applications, Vol. 2, Kluwer Academic Publishers, 1993.
1.2. The Fundamental Theorem of Algebra
The Fundamental Theorem of Algebra (FTA) asserts that a nonconstant poly-
nomial
(1.5) p(z) =
anzn
+
an−1zn−1
+ · · · + a0
with complex coefficients must vanish somewhere in the complex plane.
Eighteenth century attempts to establish this result (for polynomials with real coef-
ficients) by such worthies as Euler, Lagrange, and Laplace all proved fatally flawed;
and even the geometric proof proposed by Gauss in 1799 had a (topological) gap,
which was filled only in 1920 (by Alexander Ostrowski [O]; cf. [Sm, pp. 4-5]).
Thus, the first rigorous proof of the theorem, published by Argand in 1814, marks
an early high water mark for nineteenth century mathematics.
Complex function theory offers a particularly efficient approach for proving
FTA; and proofs using such results as Liouville’s Theorem, the Maximum
Principle, the Argument Principle, and Rouché’s Theorem appear in the standard
texts. Surprisingly, however, the simplest and shortest proof, based on the Cauchy
Integral Formula for circles, does not seem to have been recorded in the textbook
literature.
Proof of FTA. Let the polynomial p be given by (1.5), where n 1 and
an = 0. First observe that
(1.6) lim
R→∞
|p(Reiθ)|
= uniformly in θ
since
|p(z)|
|z|n(|an|
|an−1|/|z| · · ·
|a0|/|z|n)
|an|
2
|z|n
for z sufficiently large.
Now suppose that p does not vanish on C. Then q = 1/p is analytic throughout
C and q(0) = 1/p(0) = 0. By Cauchy’s integral formula,
(1.7) q(0) =
1
2πi
|z|=R
q(z)
z
dz =
1


0
q(Reiθ)dθ
for all R 0. But the integral on the right hand side of (1.7) tends to 0 by (1.6)
as R ∞, and we have the desired contradiction.
Comment. The proof given above is taken from [Z]; cf. [Sc] and the discussion
in [V].
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