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

2π

2π

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].