proof of fundamental theorem of algebra (due to Cauchy)

proof of fundamental theorem of algebra (due to Cauchy)

We will prove that any equation

f

⁢

(

z

)

:=

z

n

+

a

1

⁢

z

n

-

1

+

a

2

⁢

z

n

-

2

+

…

+

a

n

-

1

⁢

z

+

a

n

=

0

,

where the

coefficients

a

j

are

complex numbers

and

n

≧

1

,  has at least one root (

http://planetmath.org/Equation

) in

ℂ

.

Proof.

We can suppose that

a

n

≠

0

.  Denote

z

:=

x

+

i

⁢

y

where

x

,

y

are real.  Then the

function

g

⁢

(

x

,

y

)

:=

|

f

⁢

(

z

)

|

=

|

f

⁢

(

x

+

i

⁢

y

)

|

is defined and

continuous

in the whole

ℝ

2

.  Let

c

:=

∑

j

=

1

n

|

a

j

|

; it is positive.  Using the

triangle inequality

we make the estimation

|

f

⁢

(

z

)

|

=

|

z

|

n

⁢

|

1

+

a

1

z

+

a

2

z

2

+

…

+

a

n

z

n

|

≧

(

1

-

|

a

1

|

|

z

|

-

|

a

2

|

|

z

|

2

+

…

-

|

a

n

|

|

z

|

n

)

≧

(

1

-

|

a

1

|

|

z

|

-

|

a

2

|

|

z

|

+

…

-

|

a

n

|

|

z

|

)

=

|

z

|

n

⁢

(

1

-

c

|

z

|

)

≧

1

2

⁢

|

z

|

n

,

being true for

|

z

|

>

max

⁡

{

1

,

2

⁢

c

}

.  Denote

r

:=

max

⁡

{

1

,

2

⁢

c

,

2

⁢

|

a

n

|

n

}

.  Consider the disk

x

2

+

y

2

≦

r

2

.  Because it is

compact

, the function

g

⁢

(

x

,

y

)

attains at a point

(

x

0

,

y

0

)

of the disk its

absolute minimum

value (infimum) in the disk.  If

|

z

|

>

r

,  we have

g

⁢

(

x

,

y

)

=

|

f

⁢

(

z

)

|

≧

1

2

⁢

|

z

|

n

>

1

2

⁢

r

n

≧

1

2

⁢

(

2

⁢

|

a

n

|

n

)

n

=

|

a

n

|

>

0

.

Thus

g

⁢

(

x

0

,

y

0

)

≦

g

⁢

(

0

,

0

)

=

|

a

n

|

<

|

f

⁢

(

z

)

|

for

⁢

|

z

|

>

r

.

Hence

g

⁢

(

x

0

,

y

0

)

is the absolute minimum of

g

⁢

(

x

,

y

)

in the whole

complex plane

.  We show that

g

⁢

(

x

0

,

y

0

)

=

0

.  Therefore we make the antithesis that

g

⁢

(

x

0

,

y

0

)

>

0

.

Denote

z

0

:=

x

0

+

i

⁢

y

0

,

z

:=

z

0

+

u

and

f

⁢

(

z

)

=

f

⁢

(

z

0

+

u

)

:=

b

n

+

b

n

-

1

⁢

u

+

b

n

-

2

⁢

u

2

+

…

+

b

1

⁢

u

n

-

1

+

u

n

.

Then

b

n

=

f

⁢

(

z

0

)

≠

0

by the antithesis.  Moreover, denote

c

j

:=

b

j

b

n

(

j

=

1

,

2

,

…

,

n

)

,

c

0

:=

1

b

n

.

and assume that

c

n

-

1

=

c

n

-

2

=

…

=

c

n

-

k

+

1

=

0

but

c

n

-

k

≠

0

.

Thus we may write

f

⁢

(

z

)

=

b

n

⁢

(

1

+

c

n

-

k

⁢

u

k

+

c

n

-

k

-

1

⁢

u

k

+

1

+

…

+

c

0

⁢

u

n

)

.

If

c

n

-

k

=

p

⁢

(

cos

⁡

α

+

i

⁢

sin

⁡

α

)

and

u

=

ϱ

⁢

(

cos

⁡

φ

+

i

⁢

sin

⁡

φ

)

, then

c

n

-

k

⁢

u

k

=

p

⁢

ϱ

k

⁢

[

cos

⁡

(

α

+

k

⁢

φ

)

+

i

⁢

sin

⁡

(

α

+

k

⁢

φ

)

]

by

de Moivre identity

.  Choosing

ϱ

≦

1

and

φ

=

π

-

α

k

we get

c

n

-

k

⁢

u

k

=

-

p

⁢

ϱ

k

and can make the estimation

|

c

n

-

k

-

1

⁢

u

k

+

1

+

…

+

c

0

⁢

u

n

⏟

h

⁢

(

u

)

|

≦

|

c

n

-

k

-

1

|

⁢

ϱ

k

+

1

+

…

+

|

c

0

|

⁢

ϱ

n

≦

(

|

c

n

-

k

-

1

|

+

…

+

|

c

0

|

)

⁢

ϱ

k

+

1

:=

R

⁢

ϱ

k

+

1

where

R

is a constant.  Let now

ϱ

=

min

⁡

{

1

,

1

p

k

,

p

2

⁢

R

}

.  We obtain

|

f

⁢

(

z

)

|

=

|

b

n

|

⋅

|

1

-

p

⁢

ϱ

k

+

h

⁢

(

u

)

|

≦

|

b

n

|

⁢

[

|

1

-

p

⁢

ϱ

k

|

+

|

h

⁢

(

u

)

|

]

≦

|

b

n

|

⁢

[

1

-

p

⁢

ϱ

k

+

R

⁢

ϱ

k

+

1

]

≦

|

b

n

|

⁢

[

1

-

ϱ

k

⁢

(

p

-

R

⁢

ϱ

)

]

≦

|

b

n

|

⁢

[

1

-

ϱ

k

⁢

(

p

-

R

⋅

p

2

⁢

R

)

]

≦

|

b

n

|

⁢

[

1

-

1

p

⋅

p

2

]

≦

|

b

n

|

2

<

|

b

n

|

=

|

f

⁢

(

z

0

)

|

,

which result is impossible since

|

f

(

z

0

|

was the absolute minimum.  Consequently, the antithesis is wrong, and the proof is settled.

Title

proof of fundamental theorem of algebra

(due to Cauchy)

Canonical name

ProofOfFundamentalTheoremOfAlgebradueToCauchy

Date of creation

2013-03-22 19:11:10

Last modified on

2013-03-22 19:11:10

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Proof

Classification

msc 30A99

Classification

msc 12D99

Synonym

Cauchy proof of fundamental theorem of algebra

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