proof of fundamental theorem of algebra (due to d’Alembert)

proof of fundamental theorem of algebra (due to d’Alembert)

This proof, due to d’Alembert, relies on the following three facts:

•

Every

polynomial

with real

coefficients

which is of odd order has a

real root

. (This is a

corollary

of the

intermediate value theorem

.

•

Every

second order

polynomial with

complex

coefficients has two

complex roots

.

•

For every polynomial

p

with real coefficients, there exists a field

E

in which the polynomial may be factored into linear terms. (For more information, see the entry “

splitting field

”.)

Note that it suffices to prove that every polynomial with real coefficients has a complex root. Given a polynomial with complex coefficients, one can construct a polynomial with real coefficients by multiplying the polynomial by its

complex conjugate

. Any root of the resulting polynomial will either be a root of the original polynomial or the complex conjugate of a root.

The proof proceeds by

induction

. Write the degree of the polynomial as

2

n

⁢

(

2

⁢

m

+

1

)

. If

n

=

0

, then we know that it must have a real root. Next, assume that we already have shown that the

fundamental theorem of algebra

holds whenver

n

<

N

. We shall show that any polynomial of degree

2

N

⁢

(

2

⁢

m

+

1

)

has a complex root if a certain other polynomial of order

2

N

-

1

⁢

(

2

⁢

m

′

+

1

)

has a root. By our

hypothesis

, the other polynomial does have a root, hence so does the original polynomial. Hence, by induction on

n

, every polynomial with real coefficients has a complex root.

Let

p

be a polynomial of order

d

=

2

N

⁢

(

2

⁢

m

+

1

)

with real coefficients. Let its

factorization

over the

extension field

E

be

p

⁢

(

x

)

=

(

x

-

r

1

)

⁢

(

x

-

r

2

)

⁢

⋯

⁢

(

x

-

r

d

)

Next construct the

d

⁢

(

d

-

1

)

/

2

=

1

polynomials

q

k

⁢

(

x

)

=

∏

i

<

j

(

x

-

r

i

-

r

j

-

k

⁢

r

i

⁢

r

j

)

where

k

is an integer between

1

and

d

⁢

(

d

-

1

)

/

2

=

1

. Upon expanding the

product

and collecting terms, the coefficient of each power of

x

is a symmetric function of the roots

r

i

. Hence it can be expressed in terms of the coefficients of

p

, so the coefficients of

q

k

will all be real.

Note that the order of each

q

k

is

d

⁢

(

d

-

1

)

/

2

=

2

N

-

1

⁢

(

2

⁢

m

+

1

)

⁢

(

2

N

⁢

(

2

⁢

m

+

1

)

-

1

)

. Hence, by the

induction hypothesis

, each

q

k

must have a complex root. By construction, each root of

q

k

can be expressed as

r

i

+

r

j

+

k

⁢

r

i

⁢

r

j

for some choice of integers

i

and

j

. By the

pigeonhole principle

, there must exist integers

i

,

j

,

k

1

,

k

2

such that both

u

=

r

i

+

r

j

+

k

1

⁢

r

i

⁢

r

j

and

v

=

r

i

+

r

j

+

k

2

⁢

r

i

⁢

r

j

are complex. But then

r

i

and

r

j

must be complex as well. because they are roots of the polynomial

x

2

+

b

⁢

x

+

c

where

b

=

-

k

2

⁢

u

+

k

1

⁢

v

(

k

1

+

k

2

)

and

c

=

u

-

v

k

1

-

k

2

Note.

D’Alembert was an avid supporter (in fact, the co-editor) of the famous French philosophical encyclopaedia. Therefore it is a fitting tribute to have his proof appear in the web pages of this encyclopaedia.

References

1

Jean le Rond D’Alembert

: “Recherches sur le calcul intégral”.

Histoire de l’Acadḿie Royale des Sciences et Belles Lettres

, année MDCCXLVI, 182–224. Berlin (1746).

2

R. Argand

: “Réflexions sur la nouvelle théorie d’analyse”.

Annales de mathématiques

5

, 197–209 (1814).

Title

proof of fundamental theorem of algebra

(due to d’Alembert)

Canonical

name

ProofOfFundamentalTheoremOfAlgebradueToDAlembert

Date of creation

2013-03-22 14:36:06

Last modified on

2013-03-22 14:36:06

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

10

Author

rspuzio (6075)

Entry type

Proof

Classification

msc 30A99

Classification

msc 12D99

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