Number Theory: An Introduction to Mathematics

5 Complex Numbers 45

uses the result, due to Kronecker (1887), that a polynomial with coefficients from an
arbitrary fieldKdecomposes into linear factors in a fieldLwhich is a finite extension
ofK. This general result, which is not difficult to prove, is actually all that is required
for many of the previous applications of the fundamental theorem of algebra.
It is often said that the first rigorous proof of the fundamental theorem of algebra
was given by Gauss (1799). Like d’Alembert, however, Gauss assumed properties of
algebraic curves which were unknown at the time. The gaps in this proof of Gauss
were filled by Ostrowski (1920).
There are alsotopological proofsof the fundamental theorem of algebra, e.g. using
the notion of topological degree. This type of proof is intuitively appealing, but not
so easy to make rigorous. Finally, there arecomplex analysis proofs, which depend
ultimately on Cauchy’s theorem on complex line integrals. (The latter proofs are more
closely related to either the differential calculus proofs or the topological proofs than
they seem to be at first sight.)
Theexponential function ezmay be defined, for any complex value ofz,asthesum
of the everywhere convergent power series
∑

n≥ 0

zn/n!= 1 +z+z^2 /2!+z^3 /3!+···.

It is easily verified thatw(z)=ezis a solution of the differential equationdw/dz=w
satisfying the initial conditionw( 0 )=1.
For anyζ∈C, putφ(z)=eζ−zez. Differentiating by the product rule, we obtain

φ′(z)=−eζ−zez+eζ−zez= 0.

Since this holds for allz∈C,φ(z)is a constant. Thusφ(z)=φ( 0 )=eζ. Replacing
ζbyζ+z, we obtain theaddition theoremfor the exponential function:

eζez=eζ+z for allz,ζ∈C.

In particular,e−zez=1 and henceez=0foreveryz∈C.
The power series forezshows that, for any realy,e−iyis the complex conjugate
ofeiyand hence

|eiy|^2 =eiye−iy= 1.

It follows that, for all realx,y,

|ex+iy|=|ex||eiy|=ex.

Thetrigonometric functionscoszand sinzmay be defined, for any complex value
ofz, by the formulas of Euler (1740):

cosz=(eiz+e−iz)/ 2 , sinz=(eiz−e−iz)/ 2 i.

It follows at once that

eiz=cosz+isinz,

cos 0= 1 , sin 0= 0 ,

cos(−z)=cosz, sin(−z)=−sinz,