fundamental theorem of algebra 209

A function f:X →Yis said to be “one-to-one” (or

“injective”) if no two different elements of Xyield the

same output. For example, the “is the mother of” func-

tion is not one-to-one: two different people could have

the same mother. The squaring function, thought of as

a map from the set {1,2,3,4} to the set{1,4,9,16}, is one-

to-one. It is not one-to-one, however, when thought of

as a number from the set of all real numbers to the set

of all real numbers: the numbers 2 and –2, for exam-

ple, yield the same output.

A function that is both one-to-one and onto is

called a “bijection” (or sometimes a

PERMUTATION

). A

bijection f:X →Yhas the property that each element of

Y“comes from” one, and only one, element of X. Thus

it is possible to define the inverse function, denoted f–1:

Y→X, which associates to each element yof Ythe ele-

ment of Xfrom whence it came. Algebraically, if the

dependent variable yis given as an explicit formula in

terms of x, then the inverse function determines the

independent variable xas a formula in terms of y. For

example, the inverse of the function y= 3x+ 2 is given

by x= (y– 2)/3. (That is, the inverse operation of

“tripling a number and adding two” is to “subtract

two and then divide by three.”) The roles of the vari-

ables xand yhave switched, and thus the graph of the

inverse of a function can be found by switching the x-

and y-axes on the graph of the original function.

In the theory of

CARDINALITY

, bijections play a key

role in determining whether or not two sets Xand Y

have the same “size.”

See also

ALGEBRAIC NUMBER

;

HISTORY OF FUNC

-

TIONS

(essay).

fundamental theorem of algebra (d’Alembert’s the-

orem) The following important theorem in mathe-

matics is deemed fundamental to the theory of algebra:

Every polynomial p(z) = anzn+ an–1zn–1 +…+

a1z+ a0with coefficients aieither real or com-

plex numbers, an≠0, has at least one root.

That is, there is at least one complex number α

such that p(α) = 0.

By the

FACTOR THEOREM

we must then have p(z) =

(z–α)q(z) for some polynomial q(z) of degree n– 1.

Applying the fundamental theorem of algebra to the

polynomial q(z),and again to each polynomial of

degree greater than one that appears, shows that the

polynomial p(z) factors completely into n(not necessar-

ily distinct) linear factors. We have as a consequence:

Every polynomial p(z) = anzn+ an–1zn–1 +…+

a1z+ a0with coefficients aieither real or com-

plex numbers, an≠0, factors completely as

p(z) = an(z– α1)(z– α2)…(z– αn) for some

complex numbers α1, α2, …, αn.

Consequently, in the

FIELD

of complex numbers, every

polynomial of degree nhas precisely nroots (when

counted with multiplicity). For instance, the polyno-

mial z4– 2z3+ 2z2– 2z+ 1 factors as (z– 1)(z– 1)

(z–i)(z+ i) with the root 1 appearing twice. Mathe-

maticians call a field algebraically closed if every

degree-npolynomial with coefficients from that field

has precisely nroots in that field. The set of complex

numbers is thus algebraically closed. (The field of real

numbers, however, is not. The polynomial p(x) = x2+1,

for instance, does not factor within the reals.)

The fundamental theorem of algebra was first con-

jectured by Dutch mathematician Albert Girard in 1629

in his investigation of imaginary roots. C

ARL

F

RIEDRICH

G

AUSS

(1777–1855) was the first to prove the result in

his 1799 doctoral thesis. He later re-proved the result

several times throughout his life using a variety of dif-

ferent mathematical approaches, and he gave it the

name the “fundamental theorem of algebra.” In France,

the result is known as d’Alembert’s theorem to honor

the work of J

EAN

L

E

R

OND D

’A

LEMBERT

(1717–83) and

his many (unsuccessful) attempts to prove it.

To prove the theorem, it suffices to consider a com-

plex polynomial with leading coefficient equal to one:

p(z) = zn+ an–1zn–1 +…+ a1z+ a0. (Divide through by

anif necessary.) Notice that if a0= 0, then the polyno-

mial has one root, namely z= 0, and there is nothing

more to establish. Suppose then that a0is a complex

number different from zero.

Using E

ULER

’

S FORMULA

, regard the variable zas a

complex number of the form z= Rei

θ

= R(cosθ+ isinθ),

where Ris a nonzero real number and θis an angle.

Notice that as θvaries from zero to 360°, z= R(cosθ+

isinθ) traces a circle of radius Rand zn= Rnein

θ

=

Rn(cos(nθ) + isin(nθ)) wraps around a circle of radius Rn

ntimes. Notice, too, that if Ris large, then p(z)= zn+

an–1zn–1 +…+ a1z+ a0= zn

is well approximated as zn(1 + 0 +…+ 0 + 0) = zn. Thus