11 Further Remarks 77the roots of a quartic discovered by his pupil Ferrari, was the most significant Western
contribution to mathematics for more than a thousand years.
Proposition 29 still holds, but is more difficult to prove, if in its statement “has a
nonzero derivative” is replaced by “which isnot constant”. Read [57] shows that the
basic results of complex analysis may be deduced from this stronger form of Proposi-
tion 29 without the use of complex integration.
AfieldFis said to bealgebraically closedif every polynomial of positive degree
with coefficients fromFhas a root inF. Thus the ‘fundamental theorem of algebra’
says that the fieldCof complex numbers is algebraically closed. The proofs of this
theorem due to Argand–Cauchy and Euler–Lagrange–Laplace are given in Chapter 4
(by Remmert) of Ebbinghauset al.[24]. As shown on p. 77 of [24], the latter method
provides, in particular, a simple proof for the existence ofn-th roots.
Wall [72] gives a proof of the fundamental theorem of algebra, based on the notion
of topological degree, and Ahlfors [1] gives the most common complex analysis proof,
based on Liouville’s theorem that a function holomorphic in the whole complex plane
is bounded only if it is a constant. A form of Liouville’s theorem is easily deduced
from Proposition 29: if the power series
p(z)=a 0 +a 1 z+a 2 z^2 +···converges and|p(z)|is bounded for allz∈C,thenan=0foreveryn≥1.
The representation of trigonometric functions by complex exponentials appears in
§138 of Euler [25]. The various algebraic formulas involving trigonometric functions,
such as
cos 3x=4cos^3 x−3cosx,are easily established by means of this representation and the addition theorem for the
exponential function.
Some texts on complex analysis are Ahlfors [1], Caratheodory [11] and
Narasimhan [56].
The 19th century literature on quaternions is surveyed in Rothe [59]. Although
Hamilton hoped that quaternions would prove as useful as complex numbers, a quater-
nionic analysis analogous to complex analysis was first developed by Fueter (1935). A
good account is given by Sudbery [71].
One significant contribution of quaternions was indirect. After Hamilton had
shown the way, other ‘hypercomplex’ number systems were constructed, which led
eventually to the structure theory of associative algebras discussed below.
It is not difficult to show that anyautomorphismofH, i.e. any bijective map
T:H→Hsuch that
T(x+y)=Tx+Ty, T(xy)=(Tx)(Ty) for allx,y∈H,has the formTx=uxu−^1 for some quaternionuwith norm 1.
For octonions and their uses, see van der Blij [8] and Springer and Veldkamp [67].
The group of all automorphisms of the algebraOis the exceptional simple Lie group
G 2. The other four exceptional simple Lie groups are also all related toOin some way.